Download Factoring Polynomials 7 - Garnet Valley School District

Factoring Polynomials
CHAPTER
7
Solutions Key
Are you READY?
1. B; a polynomial with two terms
2. A; a whole number greater than 1 that has more
than two whole-number factors
7-1 Factors and Greatest Common
Factors
Check it OUt!
1a. 40 = 2 · 2 · 2 · 5
= ​2 ​3​· 5
3. F; a number that is multiplied by another number to
get a product
b. 33 = 3 · 11
4. C; the product of any number and a whole number
c. 49 = 7
​  ​ ​
5. E; a whole number greater than 1 that has exactly
two factors, itself and 1
d. 19 is a prime number.
6. 3, 6, 9, 12
7. 4, 8, 12, 16
8. 8, 16, 24, 32
9. 15, 30, 45, 60
10. Yes; 5 × 4 = 20
11. no; factors of 50: 1, 2, 5, 10, 25, 50
12. Yes; 8 × 15 = 120
13. Yes; 7 × 35 = 245
14. Prime
15. Prime
16. Composite; 17. Composite;
10 = 2 · 5 38 = 2 · 19
18. Composite; 19. Composite;
115 = 5 · 23 147 = 21 · 7
0. Prime
2
21. Composite;
93 = 3 · 31
22. 2(x + 5)
23. 3h(h + 1)
2(x) + 2(5) 3h(h) + 3h(1)
2
2x +10 3​h ​ ​+ 3h
24. xy​(​x ​2​- x​y 3​ ​)​
xy​(​x ​ )​ ​- xy​(x​y ​ ​)​
2
3
3
2 4
​x ​ ​y - ​x ​ ​​y ​ ​
25. 6m​(​m 2​ ​- 4m - 1)​
6m​(​m ​ ​)​- 6m​(4m)​- 6m​(1)​
6​m ​3​- 24​m 2​ ​- 6m
26. (x + 3)(x + 8)
x(x) + x(8) + 3(x) + 3(8)
​x ​2​+ 8x + 3x + 24
​x ​2​+ 11x + 24
2
27. (b - 7)(b + 1)
b(b) + b(1) - 7(b) - 7(1)
​b ​2​+ b - 7b - 7
​b 2​ ​- 6b - 7
28. (2p - 5)(p - 1)
2p(p) + 2p(-1) - 5(p) - 5(-1)
2​p ​2​- 2p - 5p + 5
2​p ​2​- 7p + 5
2
2a. Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 16: 1, 2, 4, 8, 16
The GCF of 12 and 16 is 4.
b. 15 = 3 · 5
25 = 5 · 5
The GCF of 15 and 25 is 5.
2
3a. 18​g ​ ​= 2 · 3 · 3 · g · g
27​g 3​ ​= 3 · 3 · 3 · g · g · g
The GCF of 18​g 2​ ​and 27​g 3​ ​is 9​g 2​ ​.
2
b. 16​a ​ ​= 2 · 2 · 2 · 2 · a · a
9b = 3 · 3 · b
The GCF of 16​a 2​ ​and 9b is 1.
c. 8x = 8 · x
2
7​v ​ ​= 7 · v · v
The GCF of 8x and 7​v 2​ ​is 1.
4. First find the GCF of 36 and 48 because the number
of CDs per shelves must be a common factor of 36
and 48.
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
The GCF of 36 and 48 is 12.
The greatest possible number of CDs in each shelf
is 12. Find the number of shelves if each shelf holds
12 CDs.
36 CDs by pop artists + 48 CDs by country artists
​ ________________________________________
     
   
 ​
12 CDs per shelf
= 7 shelves
Think and Discuss
1. Use a factor tree or divide the number by prime
factors until the quotient is 1.
2.
Coefficient
100
Prime factorization
of coefficient
2·2·5·5
Variable Term
x2
Variable term
as a product
x·x
100x 2
Prime
Factorization
of 100x 2
22 · 52 · x 2
29. (3n + 4)(2n + 3)
3n(2n) + 3n(3) + 4(2n) + 4(3)
6​n ​2​+ 9n + 8n + 12
6​n ​2​+ 17n + 12
223 Holt McDougal Algebra 1
Exercises
26. Factors of 14: 1, 2, 7, 14
Factors of 15: 1, 3, 5, 15
The GCF of 14 and 15 is 1.
Guided Practice
1. Possible answer: the greatest number that is a
factor of two given numbers
27. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
The GCF of 30 and 40 is 10.
2
2. 20 = 2 · 2 · 5 = 2
​  ​ ​· 5
2
2
​  ​ ​
3. 36 = 3 · 3 · 2 · 2 = 3
​  ​ ​· 2
28. 8​a 2​ ​= 2 · 2 · 2 · a · a
11 = 1 · 11
The GCF of 8
​ a 2​ ​and 11 is 1.
3
4. 27 = 3 · 3 · 3 = 3
​  ​ ​
3
5. 54 = 3 · 3 · 3 · 2 = 3
​  ​ ​· 2
29. 9s = 3 · 3 · s
3
63​s ​ ​= 3 · 3 · 7 · s · s · s
The GCF of 9s and 63​s 3​ ​is 9s.
5
6. 96 = 2 · 2 · 2 · 2 · 2 · 3 = 2
​  ​ ​· 3
7. 7
2
2
​  ​ ​
8. 100 = 2 · 2 · 5 · 5 = 2
​  ​ ​· 5
4
30. -64​n ​ ​= -1 · 2 · 2 · 2 · 2 · 2 · 2 · n · n · n · n
24​n ​2​ = 2 · 2 · 2 · 3 · n · n
The GCF of -64​n 4​ ​and 24​n 2​ ​is 8​n 2​ ​.
2
9. 75 = 3 · 5 · 5 = 3 · 5
​  ​ ​
10. Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 20, 30, 60
The GCF of 12 and 60 is 12.
31. First find the GCF of 72 and 108 because the
number of tarts must be a common factor of 72 and
108.
72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36, 72
108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
The GCF of 72 and 108 is 36, so there will be
36 fruits in each tart. There will be a total of 5 tarts:
2 raspberry and 3 blueberry.
32. 5 · t = 5t
33. 2 · 2 · x · x = 4​x ​2​
11. Factors of 14: 1, 2, 7, 14
Factors of 49: 1, 7, 49
The GCF of 14 and 49 is 7.
12. Factors of 55: 1, 5, 11, 55
Factors of 121: 1, 11, 121
The GCF of 55 and 121 is 11.
13. 6​x ​2​= 2 · 3 · x · x
5​x 2​ ​= 5 · x · x
The GCF of 6​x 2​ ​and 5​x 2​ ​is ​x 2​ ​.
34. 11
35. 2 · n = 2n
3
36. Possible answer: Even numbers greater than 2 all
have 2 as a factor and thus are not prime.
4
37. No; An odd composite number and an even
composite number can have no factors in
common other than 1.
14. 15​y ​ ​ = 3 · 5 · y · y · y
-20y = -1 · 2 · 2 · 5 · y
The GCF of 15​y 3​ ​and -20y is 5y.
15. 13​q ​ ​= 13 · q · q · q · q
2​p ​2​ = 2 · p · p
The GCF of 13​q 4​ ​and 2​p 2​ ​is 1.
16. First find the GCF of 54 and 18 because the number
of beads per necklace must be a common factor of
54 and 18.
Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
Factors of 18: 1, 2, 3, 6, 9, 18
The GCF of 54 and 18 is 18.
The greatest possible number of beads in each
necklace is 18. Find the number of necklaces if each
necklace takes 18 beads to make.
54 glass beads + 18 clay beads
​ __________________________
   
   
 ​= 4 necklaces
18 beads per necklace
Practice and Problem Solving
17. 18 = 2 · 3 · 3 = 2 · ​3 ​2​
18. 64 = 2 · 2 · 2 · 2 · 2 · 2
= ​2 ​6​
19. 12 = 2 · 2 · 3 = ​2 ​2​· 3
20. 150 = 2 · 3 · 5 · 5
=2·3·5
​  ​2​
21. 17
22. 226 = 2 · 113
38a. Since the area of a rectangle is length times width,
to find all possible whole number lengths, find
2 whole numbers that have the product 84.
1 × 84; 2 × 42; 3 × 28; 4 × 21; 6 × 14; 7 × 12
b. P = 2(7 + 12)
c. P = 2(1 + 84)
= 2(19) = 38 ft = 2(85) = 170 ft
39. First find the GCF of 35 and 40, because the
number of guards in each row must be a common
factor of 35 and 40.
35: 1, 5, 7, 35
40: 1, 2, 4, 5, 8, 10, 20, 40
The GCF of 35 and 40 is 5.
The greatest possible number of guards in each
row is 5. Find the number of rows if each row has
5 guards.
35 Cavaliers + 40 Blue Devils
​ _________________________
   
   
 ​= 15 rows
5 Guards per row
41. 8: 1, 2, 4, 8
40. 11: 1, 11
20: 1, 2, 4, 5, 10, 20
12: 1, 2, 3, 4, 6, 12
63: 1, 3, 7, 9, 21, 63
14: 1, 2, 7, 14
8 and 20; GCF = 4
12 and 14; GCF = 2
23. 49 = 7 · 7 24. 63 = 3 · 3 · 7
2
= ​3 ​2​· 7
=7
​  ​ ​
25. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 63: 1, 3, 7, 9, 21, 63
The GCF of 36 and 63 is 9.
224 Holt McDougal Algebra 1
42. 16:
1, 2, 4, 8, 16
21: 1, 3, 7, 21
27: 1, 3, 9, 27
21 and 27; GCF = 3
43. 32: 1, 2, 4, 8, 16, 32
63: 1, 3, 7, 9, 21, 63
105: 1, 3, 5, 7, 15, 21,
35, 105
63 and 105; GCF = 21
44. 25: 1, 5, 25
35: 1, 5, 7, 35
54: 1, 2, 3, 4, 6, 9, 16,18, 27, 54
25 and 35; GCF = 5
45. 35: 1, 5, 7, 35
54: 1, 2, 3, 6, 9, 18, 27, 54
72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
54 and 72; GCF = 18
46.​2 ​4​· 3; possible answer: because 48 = 2 · 24 and
24 = ​2 ​3​· 3, 48 = 2 · 2
​  ​3​· 3 = ​2 ​4​· 3
From top to bottom, left to right:
47. 36; 2; 9; 3; 2
​  ​3​· 3
​  ​2​
48. 27; 3; 9; 3
​  ​4​
49. 105; 5; 7; 2 · 3 · 5 · 7 50. 2; 14; 7; 2
​  ​3​· 7
2
3
51. 2; 2; 27; 3; 2
​  ​ ​· 3
​  ​ ​
52. 2; 34; 17; 2
​  ​3​· 17
4
53. 24; 2; 6; 3; 2
​  ​ ​· 3
54. 2; 70; 5; 2
​  ​2​· 5 · 7
3
55. 2; 2; 10; 5; 2
​  ​ ​× 5
1  ​ a​t  ​2​
56a. Use the given formula, d = vt + ​ __
2
1  ​ (2)​t  ​2​
d = (2)t + ​ __
2
2
= 2t + ​t ​ ​
b. 2t = 2 · t
2
​t ​ ​= t · t
The GCF of 2t and ​t  ​2​is t.
Test Prep
57. D; 16, 24, 48 has a GCF of 8.
58. F; the GCF of 48 and 12 is 12, and the GCF of 12
and 8 is 4.
24 ft
59.
1 ft
P = 50 ft
12 ft
2 ft
P = 28 ft
3 ft
P = 22 ft
6 ft
4 ft
P = 20 ft
Patricia should make the pen 4 ft × 6 ft because
these dimensions give the shortest perimeter and
she will need to buy the least fencing.
Challenge and Extend
60. 4​n 3​ ​ = 2 · 2 · n · n · n
16​n ​2​= 2 · 2 · 2 · 2 · n · n
8n = 2 · 2 · 2 · n
The GCF of 4​n 3​ ​, 16​n 2​ ​, and 8n is 4n.
61. 27​y ​ ​= 3 · 3 · 3 · y · y · y
18​y ​2​= 2 · 3 · 3 · y · y
81y = 3 · 3 · 3 · 3 · y
The GCF of 27​y 3​ ​, 18​y 2​ ​, and 81y is 9y.
4
63. 2​p ​ ​r = 2 · p · p · p · p · r
8​p 3​ ​​r 2​ ​ = 2 · 2 · 2 · p · p · p · r · r
16​p ​2​​r 3​ ​= 2 · 2 · 2 · 2 · p · p · r · r · r
The GCF of 2​p 4​ ​r, 8​p ​3​​r 2​ ​, and 16​p 2​ ​​r 3​ ​is 2​p 2​ ​r.
3
64. 2​x ​ ​y = 2 · x · x · x · y
8​x 2​ ​​y ​2​= 2 · 2 · 2 · x · x · y · y
17x​y 3​ ​= 17 · x · y · y · y
The GCF of 2​x 3​ ​y, 8​x 2​ ​​y ​ 2​, and 17x​y 3​ ​is xy.
4 3
65. 8​a ​ ​​b ​ ​ = 2 · 2 · 2 · a · a · a · a · b · b · b
3 3
4​a ​ ​​b ​ ​ = 2 · 2 · a · a · a · b · b · b
12​a ​2​​b ​3​= 2 · 2 · 3 · a · a · b · b · b
The GCF of 8​a 4​ ​​b ​3​, 4​a ​3​​b 3​ ​, and 12​a 2​ ​​b 3​ ​is 4​a 2​ ​​b 3​ ​.
66. 1 × 20; 2 × 10; 4 × 5; 20 × 1; 10 × 2; 5 × 4
67. Possible answer: 21, 35, 49; 7, 14, 84
68. Possible answer: 6, 35, 143
6 = 2 × 3; 35 = 5 × 7; 143 = 11 × 13
7-2 Factoring by GCF
Check it out!
1a. 5b = 2 · b
9​b 3​ ​= 3 · 3 · b · b · b
The GCF of 5b and 9​b 3​ ​is b.
5(b) + 9​b ​2​(b)
b(5 + 9​b ​2​)
 2
b. 9​d ​ ​ = 3 · 3 · d · d
​8 ​2​ = 2 · 2 · 2 · 2 · 2 · 2
The GCF of 9
​ d  2
​ ​and 8
​  ​2​is 1; it cannot be factored.
3
c. 18​y ​ ​= 2 · 3 · 3 · y · y · y
7​y ​2​ = 7 · y · y
The GCF of 18​y 3​ ​and 7​y 2​ ​is ​y 2​ ​.
-1[18y(​y ​2​) + 7(​y 2​ ​)]
-​y ​2​(18y + 7)
4
8 ft
3
62. 100 = 2 · 2 · 5 · 5
25​s 5​ ​= 5 · 5 · s · s · s · s · s
50s = 2 · 5 · 5 · s
The GCF of 100, 25​s 5​ ​, and 50s is 25.
d. 8​x ​ ​= 2 · 2 · 2 · x · x · x · x
4​x ​3​= 2 · 2 · x · x · x
2​x ​2​= 2 · x · x
The GCF of 8
​ x 4​ ​, ​4x ​3​and 2
​ x 2​ ​is ​2x 2​ ​
2
2
2
4​x ​ ​(​2x ​ ​) + 2x​(2x ​ ​) - 1(​2x 2​ ​)
2​x ​2​​(4x 2​ ​+ 2x - 1)
2
2. A = 2​x ​ ​+ 4x
= x(2x) + 2(2x)
= 2x(x + 2)
Possible expression for the dimensions of the solar
panel are 2x cm and (x + 2) cm.
3a. 4s(s + 6) - 5(s + 6)
(4s - 5)(s + 6)
b. 7x(2x + 3) + (2x + 3)
7x(2x + 3) + 1(2x + 3)
(7x + 1)(2x + 3)
c. 3x(y + 4) - 2y(x + 4)
There are no common factors.
225 Holt McDougal Algebra 1
d. 5x(5x - 2) - 2(5x - 2)
(5x - 2)(5x - 2)
(5x - 2​)2​ ​
3
2
4a. 6​b ​ ​+ 8​b ​ ​+ 9b + 12
(​6b 3​ ​+ ​8b 2​ ​) + (​ 9b + 12)​
​2b ​2(​​ 3b + 4)​+ 3​(3b + 4)​
(​2b ​2​+ 3)​(3b + 4)​
3
2
b.​4r ​ ​+ 24r + r​  ​ ​+ 6
(​4r ​3​+ 24r) + (​r 2​ ​+ 6)
4r(​r 2​ ​+ 6) + 1(​r 2​ ​+ 6)
(4r + 1)(​r 2​ ​+ 6)
2
3
5a.​15x ​ ​- ​10x ​ ​+ 8x - 12
(15​x 2​ ​- 10​x 3​ ​) + (8x - 12)
5​x ​2​(3 - 2x) + 4(2x - 3)
5​x 2​ ​(3 - 2x) + 4(-1)(2x - 3)
5​x 2​ ​(3 - 2x) - 4(3 - 2x)
(5​x 2​ ​- 4)(3 - 2x)
3. 35x = 5 · 7 · x
42= 6 · 7
The GCF of 35x and 42 is 7.
-35x + 42
-5x(7) + 6(7)
7(-5x + 6)
4. 4​x ​2​= 2 · 2 · x · x
6x = 2 · 3 · x
The GCF of 4​x 2​ ​and 6x is 2x.
-(4​x ​2​+ 6x)
-​(2x(2x) + 3(2x))​
-2x(2x + 3)
4
5. 12​h ​ ​= 2 · 2 · 3 · h · h · h · h
8​h ​2​ = 2 · 2 · 2 · h · h
6h = 2 · 3 · h
The GCF of 1
​ 2h 4​ ​, 8​h ​2​and 6h is 2h.
4
2
12​h ​ ​+ 8​h ​ ​- 6h
6​h ​3​(2h) + 4h(2h) - 3(2h)
2h(6​h 3​ ​+ 4h - 3)
2
b. 8y - 8 - x + xy
(8y - 8) + (-x + xy)
8(y - 1) - x(1 - y)
8(y - 1) - x(-1)(-1 + y)
8(y - 1) + x(y - 1)
(8 + x)(y - 1)
6.​3x ​ ​= 3 · x · x
9x = 3 · 3 · x
3 = 3
The GCF of 3​x 2​ ​, 9x and 3 is 3.
3​x ​2​- 9x + 3
​x 2​ ​(3) - 3x(3) + 1(3)
3(​x 2​ ​- 3x + 1)
Think and Discuss
7. 9​m ​ ​= 3 · 3 · m · m
m = m
The GCF of 9​m 2​ ​and m is m.
9​m ​2 ​+ m
9m(m) + 1(m)
m(9m + 1)
1. Possible answer: when you know the GCF of the
monomials in a polynomial, you can factor out the
GCF from each monomial to factor the polynomial.
2.
Factoring by GCF
1. Find the greatest common factor.
2. Write each term as a product using
the GCF.
3. Use the Distributive Property to factor
out the GCF.
4. Check by multiplying.
Exercises
Guided Practice
1. 15a = 3 · 5 · a
5​a 2​ ​= 5 · a · a
The GCF of 15a and 5​a 2​ ​is 5a.
15a - 5​a ​2​
3(5a) - a(5a)
5a(3 - a)
2. 10​g ​3​= 2 · 5 · g · g · g
3g= 3 · g
The GCF of 10​g 3​ ​and 3g is g.
10​g ​3​- 3g
10​g 2​ ​(g) - 3(g)
g(10​g 2​ ​- 3)
2
8. 14​n 3​ ​= 2 · 7 · n · n · n
7n = 7 · n
7​n ​2​ = 7 · n · n
The GCF of 14​n 3​ ​, 7n and 7​n 2​ ​is 7n.
14​n ​3​+ 7n + 7​n 2​ ​
2​n ​2​(7n) + 1(7n) + n(7n)
7n(2​n 2​ ​+ 1 + n)
9. 36f = 2 · 2 · 3 · 3 · f
2
18​f ​ ​= 2 · 3 · 3 · f · f
3 = 3
The GCF of 36f, 18​f 2​ ​and 3 is 3.
36f + 18​f ​2​+ 3
12f(3) + 6​f 2​ ​(3) + 1(3)
3(12f + 6​f ​2​+ 1)
2
10. 15​b ​ ​= 3 · 5 · b · b
7b = 7 · b
The GCF of 15​b ​2 ​and 7b is b.
-15​b 2​ ​+ 7b
-15b(b) + 7(b)
b(-15b + 7)
11. -16​t ​2​+ 320t
-t(16t) + 20(16t)
16t(-t + 20)
Using the factored form of the expression can tell
you when the rocket will land again.
226 Holt McDougal Algebra 1
12. 5(m - 2) - m(m - 2)
(5 - m)(m - 2)
Practice and Problem Solving
27. 9​y 2​ ​ = 3 · 3 · y · y
45y = 3 · 3 · 5 · y
The GCF of 9​y 2​ ​and 45y is 9y.
9​y ​2​+ 45y
y(9y) + 5(9y)
9y(y + 5)
13. 2b(b + 3) + 5(b + 3)
(2b + 5)(b + 3)
14. 4(x - 3) - x(y +2)
Cannot be factored
3
2
15.​x ​ ​+ 4​x ​ ​+ 2x + 8
(​x 3​ ​+ 4​x 2​ ​) + (​ 2x + 8)​
​x ​2​(x + 4) + 2​(x + 4)​
(​x ​2​+ 2)(x + 4)
3
2
3
2
28. 36​d 3​ ​= 2 · 2 · 3 · 3 · d · d · d
24 = 2 · 2 · 2 · 3
The GCF of 36​d 3​ ​and 24 is 12.
36​d ​2​+ 24
3​d 3​ ​(12) + 2(12)
12(3​d 3​ ​+ 2)
16. 6​x ​ ​+ 4​x ​ ​+ 3x + 2
(6​x ​3​+ 4​x 2​ ​) + (​ 3x + 2)​
2​x ​2(​​ 3x + 2)​+ 1​(3x + 2)​
(2​x ​2​+ 1)(3x + 2)
4
29. 14​x ​ ​= 2 · 7 · x · x · x · x
5​x ​2​ = 5 · x · x
The GCF of 14​x 4​ ​and 5
​ x ​2​is ​x 2​ ​.
4
2
-14​x ​ ​+ 5​x ​ ​
-14​x ​2​(​x 2​ ​) + 5(​x 2​ ​)
​x ​2​(-14​x 2​ ​+ 5)
17. 4​b ​ ​- 6​b ​ ​+10b - 15
(4​b 3​ ​- 6​b 2​ ​) + (10b - 15)
2​b ​2​(2b - 3) + 5(2b - 3)
(2​b 2​ ​+ 5)(2b - 3)
3
2
18. 2​m ​ ​+ 4​m ​ ​+ 6m + 12
(2​m ​3​+ 4​m 2​ ​) + (6m + 12)
2​m ​2​(m + 2) + 6(m + 2)
2(m + 2)(​m 2​ ​+ 3)
3
30. 15f = 3 · 5 · f
2
10​f ​ ​= 2 · 5 · f · f
The GCF of 15f and 10​f  2
​ ​is 5f.
-15f - 10​f 2​ ​
3(-5f) + 2f(-5f)
-5f(3 + 2f)
2
19. 7​r ​ ​- 35​r ​ ​+ 6r - 30
7​r ​3​- 35​r 2​ ​+ 6r - 30
7​r ​2​(r - 5) + 6(r - 5)
(7​r ​2​+ 6)(r - 5)
3
31. 4​d ​4​= 2 · 2 · d · d · d · d
​d ​3​ = d · d · d
3​d 2​ ​= 3 · d · d
The GCF of 4​d  4
​ ​, ​d  3
​ ​, and 3​d  2
​ ​is ​d  2
​ ​.
-1​(4​d 4​ ​- ​d 3​ ​+ 3​d  2
​ )​ ​
2
20. 10​a ​ ​+ 4​a ​ ​+ 5a + 2
(10​a ​3​+ 4​a 2​ ​) + (5a + 2)
2​a ​2​(5a + 2) + 1(5a + 2)
(2​a 2​ ​+ 1)(5a + 2)
2
-​d ​ ​​(​4d ​ ​- d + 3)​
2
3
2
22. 6​b ​ ​- 3b + 4 - 8b
21. 2​r ​ ​- 6r + 12 - 4r
(2​r ​2​- 6r) + (12 - 4r)
(6​b 2​ ​- 3b) + (​ 4 - 8b)​
2r(r - 3) - 4(r - 3)
3b(2b - 1) - 4(2b - 1)
(2r - 4)(r - 3)
(3b - 4)(2b - 1)
2(r - 2)(r - 3)
23. 14​q ​2​- 21q + 6 - 4q
(14​q 2​ ​- 21q) + (​ 6 - 4q)​
7q(2q - 3) - 2(2q - 3)
(7q - 2)(2q - 3)
24. 3r - ​r ​2​+ 2r - 6
3r - ​r 2​ ​+ 2r - 6
r(3 - r) - 2(3 - r)
(r - 2)(3 - r)
3
2
25. 2​m ​ ​- 6​m ​ ​+ 9 - 3m
(2​m 3​ ​- 6​m 2​ ​) + (​ 9 - 3m)​
2​m ​2​(m - 3) - 3(m - 3)
2
(2​m ​ ​- 3)(m - 3)
3
2
26. 6​a ​ ​- 9​a ​ ​- 12 + 8a
(6​a 3​ ​- 9​a 2​ ​) + (-12 + 8a)
3​a ​2​(2a - 3) + 4(2a - 3)
2
32. 14​x ​ ​= 2 · 7 · x · x · x
63​x 2​ ​= 3 · 3 · 7 · x · x
7x = 7 · x
The GCF of 14​x 2​ ​, 63​x ​2​and 7x is 7x.
14​x ​3​+ 63​x 2​ ​- 7x
2​x ​2​(7x) + 9x(7x) - 1(7x)
7x(2​x 2​ ​+ 9x - 1)
2
33. 21​c ​ ​= 3 · 7 · c · c
14c = 2 · 7 · c
The GCF of 21​c 2​ ​and 14c is 7c.
21​c ​2 ​+ 14c
3c(7c) + 2(7c)
7c(3c + 2)
34. 33​d ​3​= 3 · 11 · d · d · d
22d = 2 · 11 · d
11 = 11
The GCF of 3
​ 3d 3​ ​, 22d and 11 is 11.
33​d ​3​+ 22d + 11
3​d 3​ ​(11) + 2d(11) + 1(11)
11(3​d ​3​+ 2d + 1)
2
(3​a ​ ​+ 4)(2a - 3)
227 Holt McDougal Algebra 1
35. 5​g 3​ ​ = 5 · g · g · g
15​g 2​ ​= 3 · 5 · g · g
The GCF of 5​g 3​ ​and 15​g 2​ ​is 5​g 2​ ​.
-5​g ​3​- 15​g 2​ ​
g(-5​g ​2​) + 3(-5​g 2​ ​)
-5​g ​2​(g + 3)
36. S = P + Prt
= 1(P) + rt(P)
= P(1 + rt)
P(1 + rt) is the factored expression for P + Prt.
37. Cannot be factored
38. -4x(x + 2) + 9(x + 2)
(x + 2)(-4x + 9)
39. 6y(y - 7) + (y - 7)
6y(y - 7) + 1(y - 7)
(6y + 1)(y - 7)
40. a(x - 3) + 2b(x - 3)
(a + 2b)(x - 3)
41. -3(2 + b) + 4b(b + 2) 42. 5(3x - 2) + x(3x - 2)
(-3 + 4b)(b + 2)
(5 + x)(3x - 2)
43.2​a ​3​- 8​a 2​ ​+ 3a - 12
(2​a 3​ ​- 8​a 2​ ​) + (3a - 12)
2​a ​2​(a - 4) + 3(a - 4)
(2​a 2​ ​+ 3)(a - 4)
3
2
44.​x ​ ​+ 3​x ​ ​+ 5x + 15
(​x 3​ ​+ 3​x 2​ ​) + (5x + 15)
​x ​2​(x + 3) + 5(x + 3)
(​x 2​ ​+ 5)(x + 3)
3
2
45.6​x ​ ​+ 18​x ​ ​+ x + 3
(6​x 3​ ​+ 18​x 2​ ​) + (x + 3)
6​x ​2​(x + 3) + 1(x + 3)
(6​x 2​ ​+ 1)(x + 3)
3
2
46.7​x ​ ​+ 2​x ​ ​+ 28x + 8
3
(7​x ​ ​+ 2​x 2​ ​) + (28x + 8)
​x ​2​(7x + 2) + 4(7x + 2)
(​x ​2​+ 4)(7x + 2)
3
2
47.​n ​ ​- 2​n ​ ​+ 5n - 10
3
(​n ​ ​- 2​n 2​ ​) + (5n - 10)
​n ​2​(n - 2) + 5(n - 2)
(​n 2​ ​+ 5)(n - 2)
3
2
48.10​b ​ ​- 16​b ​ ​+ 25b - 40
3
2
(10​b ​ ​- 6​b ​ ​) + (25b - 40)
2​b ​2​(5b - 8) + 5(5b - 8)
(2​b 2​ ​+ 5)(5b - 8)
3
2
49. 2​m ​ ​- 2​m ​ ​+ 3 - 3m
(2​m 3​ ​- 2​m 2​ ​) - (3m - 3)
2​m ​2​(m - 1) - 3(m - 1)
(2​m ​2​- 3)(m - 1)
3
2
50.2​d ​ ​- ​d ​ ​- 3 + 6d
3
(2​d ​ ​- ​d 2​ ​) + (6d - 3)
​d ​2​(2d - 1) + 3(2d -1)
(​d 2​ ​+ 3)(2d - 1)
3
2
51.6​f ​ ​- 8​f ​ ​+ 20 - 15f
3
(6​f ​ ​- 8​f  2
​ ​) - (15f - 20)
2​f  2
​ ​(3f - 4) - 5(3f - 4)
(2​f ​ 2​- 5)(3f - 4)
2
3
52.5​k ​ ​- ​k ​ ​+ 3k - 15
(5​k 2​ ​- ​k 3​ ​) + (3k - 15)
​k ​2​(5 - k) + 3(k - 5)
​k 2​ ​(5 - k) - 3(5 - k)
(​k 2​ ​- 3)(5 - k)
3
2
53.​b ​ ​- 2b - 8 + 4​b ​ ​
(​b ​3​+ 4​b 2​ ​) - (2b + 8)
​b ​2​(b + 4) - 2(b + 4)
(​b 2​ ​- 2)(b + 4)
2
54.20 - 15x - 6​x ​ ​+ 8x
(20 - 15x) + (8x - 6​x 2​ ​)
5(4 - 3x) + 2x(4 - 3x)
(5 + 2x)(4 - 3x)
55. Given GCF of 4v:
16v + 12​v ​2​
4(4v) + 3v(4v)
4v(4 + 3v)
56. Given GCF of 5x:
15x - 25​x ​2​
5x(3) - 5x(5x)
5x(3 - 5x)
57. Given GCF of -8​k 2​ ​:
-16​k ​3​- 24​k 2​ ​
-8​k ​2​(2k) - 8​k 2​ ​(3)
-8​k ​2​(2k + 3)
58. Given GCF of 1:
-x - 10
-1(x) - 1(10)
-1(x + 10)
59.​x ​2​+ 5x; polynomial has 2 terms, so it is a binomial.
​x 2​ ​+ 5x
x(x) + 5(x)
x(x + 5)
2
60. 28​c ​ ​- 49c; polynomial has 2 terms, so it is a
binomial.
28​c 2​ ​- 49c
4c(7c) - 7(7c)
7c(4c - 7)
61.​a ​4​+ ​a 3​ ​+ a
​  2​ ​; polynomial has 3 terms, so it is a
trinomial.
​a ​4​+ ​a 3​ ​+ ​a 2​ ​
​a ​2​(​a 2​ ​) + a(​a 2​ ​) + 1(​a 2​ ​)
​a ​2​(​a 2​ ​+ a + 1)
2
3
62. 36 + 99r - 40​r ​ ​- 110​r ​ ​; polynomial has 4 terms,
so it is a polynomial.
36 + 99r - 40​r 2​ ​- 110​r ​3​
(36 + 99r) - (40​r 2​ ​+ 110​r ​3​)
9(4 + 11r) - 10​r ​2​(4 + 11r)
(11r + 4)(-10​r ​2​+ 9)
63a. Let x be the interest rate of the CDs that Justin’s
aunt purchased for him.
For CDs purchased in 2004, n = 3 and P = 100.
The value of the CDs purchased in 2004 is 1
​ 00x ​3​.
For CDs purchased in 2005, n = 2 and P = 200.
The value of the CDs purchased in 2005 is 2
​ 00x 2​ ​.
For CDs purchased in 2005, n = 2 and P = 400.
The value of the CDs purchased in 2006 is 400x.
228 Holt McDougal Algebra 1
b. The total value is 100​x ​3​+ 200​x 2​ ​+ 400x + 800.
3
Challenge and Extend
74. 6a​b 2​ ​- 24​a 2​ ​
6a(​b 2​ ​) - 6a(4a)
6a(​b 2​ ​- 4a)
2
c. 100​x ​ ​+ 200​x ​ ​+ 400x + 800
(100​x ​3​+ 200​x 2​ ​) + (400x + 800)
100​x ​2(​ x + 2) + 400(x + 2)
(100​x ​2​+ 400)(x + 2)
100(​x 2​ ​+ 4)(x + 2)
100(​1.09 ​2​+ 4)(1.09 + 2) = 1603.1229; $1603.12
64. The area of the figure is the sum of the areas of the
rectangle and the triangle. The area of the rectangle
is 2x(2x + 6), and the area of the triangle is
1 ​   · 2x(x + 8). The sum is
​ __
2
1 ​   · 2x(x + 8).
2x(2x + 6) + ​ __
2
1 ​   · 2x(x + 8)
2x(2x + 6) + ​ __
2
2
2
4​x ​ ​+ 12x + ​x ​ ​+ 8x
5​x ​2​+ 20x
5x(x + 4)
Method 2
(3a - 4a) - (3b - 4b)
a(3 - 4) - b(3 - 4)
(a - b)(3 - 4)
(a - b)(-1)
(b - a)
65. Method 1
(3a - 3b) - (4a - 4b)
3(a - b) - 4(a - b)
(3 - 4)(a - b)
-1(a - b)
(b - a)
1 ​  (​x 3​ ​- 2x + 2​x 2​ ​- 4)
66.​ __
2
1  ​ [(​x 3​ ​- 2x) + (2​x ​2​- 4)]
​ __
2
1  ​ [x(​x 2​ ​- 2) + 2(​x 2​ ​- 2)]
​ __
2
1  ​ (​x 2​ ​- 2)(x + 2)
​ __
2
The base of the triangle is ​x 2​ ​- 2.
67. If the sum of two binomials is 0, they are
opposite binomials.
68a. Either a or b, or both must equal 0.
b. The product of t and (3 - t) is 0, so at least one of
the factors must be 0.
c. The two times are t = 0 and t = 3.
69a. Commutative Property of Addition
b. Association Property of Addition
c. Distribution Property
d. Distribution Property
2
2
70. A is incorrect because ​n ​ ​≠ ​n ​ ​(0).
Test Prep
71. D;
72. G;
3
2
24​x ​ ​- 12​x ​ ​
12​x ​2​(2x) - 12​x 2​ ​(1)
12​x ​2​(2x - 1)
73. C;
2
18​x ​ ​+ 36x
18x(x) + 18x(2)
18x(x + 2)
75. -72​a 2​ ​​b 2​ ​- 45ab
-9ab(8ab) - 9ab(5)
-9ab(8ab + 5)
76. -18​a ​2​​b 2​ ​+ 21ab
77. ab + bc + ad + cd
-3ab(6ab) - 3ab(-7)
(ab + bc) + (ad + cd)
-3ab(6ab - 7)
b(a + c) + d(a + c)
(b + d)(a + c)
79.​x ​3​- 4​x 2​ ​+ 3x -12
78. 4​y ​2​+ 8ay - y -2a
2
(4​y ​ ​+ 8ay) - (y + 2a)
(​x 3​ ​- 4​x 2​ ​) + (3x - 12)
4y(y + 2a) - 1(y + 2a)
​x ​2​(x - 4) + 3(x - 4)
(4y - 1)(y + 2a)
(​x 2​ ​+ 3)(x - 4)
2
2
0a. A = π​R ​ ​- π​r ​ ​
8
A = π(​R ​2​) - π(​r 2​ ​)
A = π(​R ​2​- ​r 2​ ​)
2
2
2
b. A = π(​12 ​ ​- ​5 ​ ​) = π(119) ≈ 374​cm ​ ​
7-3 factoring ​x 2​ ​+ bx + c
Check it out!
a.​x 2​ ​+ 10x + 24
1
(x + 1)(x + 24)= ​x 2​ ​+ 25x + 24 7
(x + 2)(x + 12)= ​x ​2​+ 14x + 24 7
(x + 3)(x + 8) = ​x ​2​+ 11x + 24 7
(x + 4)(x + 6) = ​x ​2​+ 10x + 24 3
The factors of x​  ​2​+ 10x + 24 are (x + 4) and
(x + 6).
​x ​2​+ 10x + 24 = (x + 4)(x + 6)
2
b.​x ​ ​+ 7x + 12
(x + 1)(x + 12)= ​x 2​ ​+ 13x + 12 7
(x + 2)(x + 6) = ​x ​2​+ 8x + 12 7
(x + 3)(x + 4) = ​x ​2​+ 7x + 12 3
The factors of x​  2​ ​+ 7x + 12 are (x + 3) and
(x + 4).
​x ​2​+ 7x + 12 = (x + 3)(x + 4)
2
2a.​x ​ ​+ 8x + 12
​________________
Factors
of 12   
Sum​
1 and 12
13 7
2 and 6 8 3
(x + 2)(x + 6)
b.​x 2​ ​- 5x + 6
​Factors
of 6   
Sum​
_______________
-1 and -6 -7 7
-2 and -3 -5 3
(x - 2)(x - 3)
c.​x 2​ ​+ 13x + 42
​Factors
​________________
of 42   
Sum​​
1 and 42 43 7
2 and 21 23 7
3 and 14 17 7
6 and 7 13 3
(x + 6)(x + 7)
2
x​  ​ ​+ 3x - 6x - 18
(​x ​2​+ 3x) - (6x + 18)
x(x + 3) - 6(x + 3)
(x - 6)(x + 3)
229 Holt McDougal Algebra 1
d.​x 2​ ​- 13x + 40
​Factors
of 40   
Sum​
________________
-1 and -40 -41 7
-2 and -20 -22 7
-4 and -10 -14 7
-5 and -8 -13 3
(x - 5)(x - 8)
a.​x 2​ ​+ 2x - 15
3
​Factors
​__________________
of -15  
Sum​​
-1 and 15
14 7
-3 and 5 2 3
(x - 3)(x + 5)
b.​x 2​ ​- 6x + 8
​Factors
of 8   
Sum​
_______________
-1 and -8 -9 7
-2 and -4 -6 3
(x - 2)(x - 4)
c.​x ​2​- 8x - 20
​Factors
​_
of -20  
Sum​​
__________________
_________________
1 and -20 -19 7
2 and -10 -8 3
(x -10)(x + 2)
4.​n ​2​- 7n + 10
​ _______________
Factors
of 10   
Sum​
_
-1 and -10 -11 7
-2 and -5 -7 3
(n - 5)(n - 2)
n
​n ​2​- 7n + 10
0
​0 ​ ​- 7(0) + 10 = 10
1
​1 ​ ​- 7(1) + 10 = 4
2
​2 ​ ​- 7(2) + 10 = 0
3
​3 ​ ​- 7(3) + 10 = -2
4
​4 ​ ​- 7(4) + 10 = -2
n
(n - 5)(n - 2)
2
2
2
2
2
0
(0 - 5)(0 - 2) = 10
1
(1 - 5)(1 - 2) = 4
2
(2 - 5)(2 - 2) = 0
3
(3 - 5)(3 - 2) = -2
4
(4 - 5)(4 - 2) = -2
Think and discuss
1. Find the 2 factors of 14 that have a sum of 9: 2 and
7. Then use these numbers as the constants in the
factors: (x + 2)(x + 7).
Check (x + 2)(x + 7) = x​  2​ ​+ 2x + 7x + 14
= ​x ​2​+ 9x + 14 3
2
2. For ​x ​ ​+ bx + c = (x + m)(x + n), if c > 0 and
b > 0, m > 0 and n > 0. If c > 0 and b < 0, m < 0
and n < 0. If c < 0 and b > 0, the greater of m and
n is positive and the lesser is negative. If c < 0 and
b < 0, the greater of m and n is negative and the
lesser is positive.
3.
Factoring
x 2 + bx + c
c is positive,
and b is positive.
x 2 + 5x + 6
(x + 2)(x + 3)
c is negative,
and b is positive.
x2 + x - 6
(x - 2)(x + 3)
c is positive,
and b is negative.
x 2 - 5x + 6
(x - 2)(x - 3)
c is negative,
and b is negative.
x2 - x - 6
(x + 2)(x - 3)
Exercises
Guided Practice
1.​x 2​ ​+ 13x + 36
(x + 1)(x + 36) = x​  2​ ​+ 37x + 36 7
(x + 2)(x + 18) = x​  2​ ​+ 20x + 36 7
(x + 3)(x + 12) = x​  2​ ​+ 15x + 36 7
(x + 4)(x + 9) = ​x ​2​+ 13x + 36 3
The factors of x​  2​ ​+ 13x + 36 are (x + 4) and
(x + 9).
​x ​2​+ 13x + 36 = (x + 4)(x + 9)
2
2.​x ​ ​+ 11x + 24
(x + 1)(x + 24) = x​  2​ ​+ 25x + 24 7
(x + 2)(x + 12) = x​  2​ ​+ 14x + 24 7
(x + 3)(x + 8) = ​x ​2​+ 11x + 24 3
The factors of x​  2​ ​+ 11x + 24 are (x + 3) and
(x + 8).
​x ​2​+ 11x + 24 = (x + 3)(x + 8)
2
3.​x ​ ​+ 14x + 40
(x + 1)(x + 40) = x​  2​ ​+ 41x + 40 7
(x + 2)(x + 20) = x​  2​ ​+ 22x + 40 7
(x + 4)(x + 10) = x​  2​ ​+ 14x + 40 3
The factors of x​  ​2​+ 14x + 40 are (x + 4) and
(x + 10).
​x ​2​+ 14x + 40 = (x + 4)(x + 10)
2
4.​x ​ ​+ 4x + 3
​_______________
Factors
of 3   
Sum​
1 and 3 4 3
(x + 1)(x + 3)
2
5.​x ​ ​+ 10x + 16
​________________
Factors
of 16   
Sum​
1 and 16
17 7
2 and 8 10 3
(x + 2)(x + 8)
6.​x ​2​+ 15x + 44
​________________
Factors
of 44   
Sum​
1 and 44 45 7
2 and 22 24 7
4 and 11 15 3
(x + 4)(x + 11)
7.​x 2​ ​- 7x + 6
​_______________
Factors
of 6   
Sum​
-1 and -6 -7 3
(x - 1)(x - 6)
2
8.​x ​ ​- 9x + 14
​________________
Factors
of 14   
Sum​
-1 and -14 -15 7
-2 and -7 -9 3
(x - 2)(x - 7)
230 Holt McDougal Algebra 1
9.​x 2​ ​- 11x + 24
​Factors
of 24   
Sum​
________________
-1 and -24 -25 7
-2 and -12 -14 7
-3 and -8 -11 3
-4 and -6 -10 3
(x - 3)(x - 8)
10.​x ​2​+ 6x -7
​Factors
of -7   
Sum​​
_________________
-1 and 7 6 3
(x - 7)(x + 1)
2
11.​x ​ ​+ 6x - 27
​__________________
Factors
of -27  
Sum​
-1 and 27 26 7
-3 and 9 6 3
(x - 3)(x + 9)
practice and problem solving
17.​x 2​ ​+ 13x + 30
(x + 1)(x + 30) = x​  2​ ​+ 31x + 30 7
(x + 2)(x + 15) = x​  2​ ​+ 17x + 30 7
(x + 3)(x + 10) = x​  2​ ​+ 13x + 30 3
The factors of x​  ​2​+ 13x + 30 are (x + 3) and
(x + 10).
​x ​2​+ 13x + 30 = (x + 3)(x + 10)
2
18.​x ​ ​+ 11x + 28
(x + 1)(x + 28) = x​  2​ ​+ 29x + 28 7
(x + 2)(x + 14) = x​  2​ ​+ 16x + 28 7
(x + 4)(x + 7) =
​x 2​ ​+ 11x + 28 3
2
The factors of x​  ​ ​+ 11x + 28 are (x + 4) and
(x + 7).
​x ​2​+ 11x + 28 = (x + 4)(x + 7)
2
12.​x ​ ​+ x - 30
​__________________
Factors
of -30  
Sum​
-1 and 30 29 7
-2 and 15 13 7
-3 and 10 7 7
-5 and 6 1 3
(x - 5)(x + 6)
19.​x ​ ​+ 16x + 48
(x + 1)(x + 48) = x​  2​ ​+ 49x + 48 7
(x + 2)(x + 24) = x​  2​ ​+ 26x + 48 7
(x + 3)(x + 16) = x​  2​ ​+ 19x + 48 7
(x + 4)(x + 12) = x​  2​ ​+ 16x + 48 3
The factors of x​  ​2​+ 16x + 48 are (x + 4) and
(x + 12).
​x ​2​+ 16x + 48 = (x + 4)(x + 12)
13.​x ​2​- x - 2
​Factors
of -2   
Sum​
_________________
1 and -2 -1 3
(x + 1)(x - 2)
20.​x ​ ​+ 12x + 11
​_
Factors
of 11   
Sum​
_______________
1 and 11 12 3
(x + 1)(x + 11)
2
2
14.​x ​ ​- 3x - 18
​__________________
Factors
of -18   Sum​​
  
1 and -17
-16 7
2 and -9 -7 7
3 and -6 -3 3
(x + 3)(x - 6)
2
15.​x ​ ​- 4x - 45
​__________________
Factors
of -45   Sum​​
  
1 and -45 -44 7
3 and -15 -12 7
5 and -9 -4 3
(x + 5)(x - 9)
2
16.​n ​ ​+ 6n - 7
​_________________
Factors
of -7   
Sum​
-1 and 7 6 3
(n - 1)(n + 7)
2
​n ​ ​+ 6n - 7
n
0
2
​0 ​ ​+ 6(0) - 7 = -7
2
1 ​1 ​ ​+ 6(1) - 7 = 0
2
2 ​2 ​ ​+ 6(2) - 7 = 9
2
3 ​3 ​ ​+ 6(3) - 7 = 20
2
4 ​4 ​ ​+ 6(4) - 7 = 33
n
(n - 1)(n + 7)
0 (0 - 1)(0 + 7) = -7
2
2
21.​x ​ ​+ 16x + 28
​________________
Factors
of 28   
Sum​
1 and 28 29 7
2 and 14 16 3
(x + 2)(x + 14)
22.​x ​2​+ 15x + 36
​________________
Factors
of 36   
Sum​
1 and 36 37 7
2 and 18 20 7
3 and 12 15 3
(x + 3)(x + 12)
23.​x ​2​- 6x + 5
​_______________
Factors
of 5   
Sum​
-1 and -5 -6 3
(x - 1)(x - 5)
2
24.​x ​ ​- 9x + 18
​________________
Factors
of 18   
Sum​
-1 and -18 -19 7
-2 and -9 -11 7
-3 and -6 -9 3
(x - 3)(x - 6)
25.​x ​2​- 12x + 32
​________________
Factors
of 32   
Sum​
-1 and -32 -33 7
-2 and -16 -18 7
-4 and -8 -12 3
(x - 4)(x - 8)
1 (1 - 1)(1 + 7) = 0
2 (2 - 1)(2 + 7) = 9
3 (3 - 1)(3 + 7) = 20
4 (4 - 1)(4 + 7) = 33
231 Holt McDougal Algebra 1
26.​x 2​ ​+ x - 12
​__________________
Factors
of -12  
Sum​
-1 and 12 11 7
-2 and 6 4 7
-3 and 4 1 3
(x - 3)(x + 4)
27.​x 2​ ​+ 4x - 21
​__________________
Factors
of -21  
Sum​
-1 and 21 20 7
-3 and 7 4 3
(x - 3)(x + 7)
28.​x ​2​+ 9x - 36
​__________________
Factors
of -36  
Sum​
-1 and 36 35 7
-2 and 18 16 7
-3 and 12 9 3
(x - 3)(x + 12)
29.​x ​2​- 12x - 13
​__________________
Factors
of -13  
Sum​
1 and -13 -12 3
(x + 1)(x - 13)
2
30.​x ​ ​- 10x - 24
​__________________
Factors
of -24  
Sum​
1 and -24 -23 7
2 and -12 -10 3
(x + 2)(x - 12)
31.​x ​2​- 2x - 35
​__________________
Factors
of -35  
Sum​
1 and -35 -33 7
5 and -7 -2 3
(x + 5)(x - 7)
32.​n ​2​- 12n - 45
​ _________________
Factors
of -45  
Sum​
_
1 and -45 -44 7
3 and -15 -12 3
(n + 3)(n - 15)
n
​n ​2​- 12n - 45
0
​0 ​ ​- 12(0) - 45 = -45
1
​1 ​ ​- 12(1) - 45 = -56
2
2
2
2
​2 ​ ​- 12(2) - 45 = -65
3
​3 ​ ​- 12(3) - 45 = -72
4
​4 ​ ​- 12(4) - 45 = -77
2
2
n
(n + 3)(n - 15)
0
(0 + 3)(0 - 15) = -45
1
(1 + 3)(1 - 15) = -56
2
(2 + 3)(2 - 15) = -65
3
(3 + 3)(3 - 15) = -72
4
(4 + 3)(4 - 15) = -77
34. A;
33. C;
2
​x 2​ ​- 7x + 10
​x ​ ​+ 3x - 10
​__________________
​________________
Factors
of -10  
Sum​
Factors
of 10   
Sum​
-1 and 10 -1 and -10 -11 7
9 7
-2 and 5 3 3
-2 and -5 -7 3
(x - 2)(x + 5)
(x - 2)(x - 5)
35. D;
36. B;
​x 2​ ​- 9x - 10
​x 2​ ​+ 11x + 10
​__________________
​________________
Factors
of -10  
Sum​
Factors
of 10   
Sum​
1 and -10 1 and 10 -9 3
11 3
(x + 1)(x - 10)
(x + 1)(x + 10)
37. They are inverse operations.
2
38.​x ​ ​+ x - 20
​__________________
Factors
of -20  
Sum​
-1 and 20 19 7
-2 and 10 8 7
-4 and 5 1 3
(x - 4)(x + 5)
39.​x ​2​- 11x + 18
​________________
Factors
of 18   
Sum​
-1 and -18 -19 7
-2 and -9 -11 3
(x - 2)(x - 9)
40.​x ​2​- 4x - 21
​__________________
Factors
of -21  
Sum​
1 and -21 -20 7
3 and -7 -4 3
(x + 3)(x - 7)
41.​x ​2​+ 10x + 9
​_______________
Factors
of 9   
Sum​
1 and 9 10 3
(x + 1)(x + 9)
2
42.​x ​ ​- 12x + 32
​__________________
Factors
of +32  
Sum​
-1 and -32 -33 7
-2 and -16
-18 7
-4 and -8 -12 3
​x ​2​- 12x + 32 = (x - 4)(x - 8).
2
43.​x ​ ​+ 13x + 42
​________________
Factors
of 42   
Sum​
1 and 42 43
2 and 21 23
3 and 14 17
6 and 7 13
(x + 6)(x + 7)
7
7
7
3
44.​x ​2​- 7x + 12
​________________
Factors
of 12   
Sum​
-1 and -12 -13 7
-2 and -6 -8 7
-3 and -4 -7 3
(x - 3)(x - 4)
45.​x ​2​+ 11x + 18
​________________
Factors
of 18   
Sum​
1 and 18 19 7
2 and 9 11 3
(x + 2)(x + 9)
232 Holt McDougal Algebra 1
46.​x 2​ ​- 6x - 27
​ _________________
Factors
of -27  
Sum​
_
1 and -27 -26 7
3 and -9 -6 3
(x + 3)(x - 9)
47.​x ​2​+ 5x - 24
​__________________
Factors
of -24  
Sum​
-1 and 24 23 7
-2 and 12 10 7
-3 and 8 5 3
(x - 3)(x + 8)
48.​x ​2​- 10x + 21
​Factors
of 21   
Sum​
________________
-1 and -21 -22 7
-3 and -7 -10 3
(x - 3)(x - 7)
2
49.​x ​ ​+ 4x - 45
​__________________
Factors
of -45  
Sum​
-1 and 45 44 7
-3 and 15 12 7
-5 and 9 4 3
(x - 5)(x + 9)
50.​n ​2​+ 11n + 28
​________________
Factors
of 28   
Sum​
1 and 28 29 7
2 and 14 16 7
4 and 7 11 3
(n + 4)(n + 7)
​n ​2​+ 11n + 28
n
58. Sign of c: negative
Binomial Factors: (x - 1)(x + 3)
Signs of Numbers in Binomials: negative; positive
59. Sign of c: negative
Binomial Factors: (x + 1)(x - 3)
Signs of Numbers in Binomials: positive; negative
60.​x 2​ ​+ 6x + 8 = x​  2​ ​+ 2x + 4x + 2 · 4
= (x + 2)(x + 4)
The width of the rectangle is (x + 2).
2 ≠ 4 so (x + 2) ≠ (x + 4)
The rectangle cannot be a square.
61a. v = 0, a = 2
1  ​ a​t 2​ ​
d = vt + ​ __
2
1 ​  (2)​t 2​ ​
= (0)t + ​ __
2
2
= ​t ​ ​
2
c.​t ​ ​- 4t
t(t) - 4(t)
t(t - 4)
b. v = 4, a = 0
1  ​ at​ 2​ ​
d = vt + ​ __
2
1  ​ (0)​t 2​ ​
= (4)t + ​ __
2
= 4t
63. True
2
64. False; the correct factorization is (x - 1)(x + 2).
2
65. False; the correct factorization is (x - 4)(x + 1).
​1 ​ ​+ 11(1) + 28 = 40
2
​2 ​ ​+ 11(2) + 28 = 54
3
​3 ​ ​+ 11(3) + 28 = 70
​4 ​ ​+ 11(4) + 28 = 88
(n + 4)(n + 7)
0 (0 + 4)(0 + 7) = 28
1 (1 + 4)(1 + 7) = 40
2 (2 + 4)(2 + 7) = 54
3 (3 + 4)(3 + 7) = 70
4 (4 + 4)(4 + 7) = 88
51. Approximately 1.5 yards
2
52.​x ​ ​+ 8x + 12 = ​x ​ ​+ 2x + 6x + 2 · 6
= (x + 2)(x + 6)
The width of the rectangle is (x + 2) ft.
2
57. Sign of c: positive
Binomial Factors: (x - 1)(x - 3)
Signs of Numbers in Binomials: both negative
2
​0 ​ ​+ 11(0) + 28 = 28
1
2
56.​x 2​ ​+ 2x - 8
(x + 4)(x - 2)
2
0
n
2
55.​x ​ ​+ 6x + 8
​x 2​ ​+ 2x + 4x + 2 · 4
(x + 2)(x + 4)
62.​x 2​ ​+ 9x + 14 = x​  2​ ​+ 2x + 7x + 2 · 7
= (x + 2)(x + 7)
The width of the platform is (x + 2) ft.
2
4
54.​x 2​ ​+ 5x + 6
​x 2​ ​+ 2x + 3x + 2 · 3
(x + 2)(x + 3)
2
3a.​x ​ ​+ 3x + 2 = ​x ​ ​+ x + 2x + 1 · 2
5
= (x + 1)(x + 2)
Length: (x + 2) ft
Width: (x + 1) ft
66. False; many trinomials cannot be factored,
2
e.g. x​  ​ ​- 12x - 32.
2
2
67.​x ​ ​- 6x + 8
​x ​2​- 2x - 4x + 2 · 4
(x - 2)(x - 4)
68.​x ​ ​- 2x - 8
​x ​2​+ 2x - 4x - 2 · 4
(x + 2)(x - 4)
69.​x ​2​+ 2x - 8
​x ​2​- 2x + 4x - 2 · 4
(x - 2)(x + 4)
70.​x ​ ​+ 6x + 8
​x ​2​+ 2x + 4x + 2 · 4
(x + 2)(x + 4)
2
1a.​x ​2​+ 12x + 20 = x​  2​ ​+ 2x + 10x + 2 · 10
7
= (x + 2)(x + 10)
The length of the fountain is (x + 10) ft.
b. Length: (x + 10) + 2 · 2 = (x + 14) ft
Width: (x + 2) + 2 · 2 = (x + 6) ft
2
c. (x + 14)(x + 6) = x​  ​ ​+ 14x + 6x + 84
= ​x 2​ ​+ 20x + 84
The total area covered is (​x ​2 ​+ 20x + 84) ​ft ​2​.
b.​x ​2​+ 8x + 15 = ​x 2​ ​+ 3x + 5x + 3 · 5
= (x + 3)(x + 5)
Length: (x + 3) ft
Width: (x + 5) ft
c. The length will increase by 1 ft. The width will
increase by 4 ft.
233 Holt McDougal Algebra 1
72.​x 2​ ​+ bx + 6
​_______________
Factors
of 6   
Sum​
1 and 6 7
2 and 3 5
-2 and -3 -5
-1 and -6 -7
Possible values of b are 7, 5, -5, and -7.
83.​x 2​ ​+ bx + 28
​________________
Factors
of 28   
Sum​
1 and 28 29
2 and 14 16
4 and 7 11
Possible values of b are 29, 16 and 11.
84.​x ​2 ​+ bx + 32
​ _______________
Factors
of 32   
Sum​
_
-1 and -32 -33
-2 and -16 -18
-4 and -8 -12
Possible values of b are -33, -18 and -12.
test prep
3. D; (x + 2)(x - 12) = x​  2​ ​+ 2x - 12x - 24
7
= ​x 2​ ​- 10x - 24
74. H;
2
​x ​ ​+ bx - 20
​__________________
Factors
of -20  
Sum​
-1 and 20 19
-2 and 10 8
-4 and 5 1
Possible values of b are 19, 8, and 1.
5a.​x ​2​+ 13x + 42 = x​  2​ ​+ 6x + 7x + 6 · 7
8
= (x + 6)(x + 7)
The length of the garden is (x + 7) ft.
b. 2 · [(x + 6) + (x + 7)] = 2 · (2x + 13) = 4x + 26
The perimeter is (4x + 26) ft.
c. 2 · [4(5) + 26] = 2 · 46 = 92.00
The cost to fence the garden is $92.00
75. C;
​x 2​ ​+ bx - 36
​__________________
Factors
of -36  
Sum​
-1 and 36 35
-2 and 18 16
-3 and 12 9
-4 and 9 5
-6 and 6 0
Possible values of b are 35, 16, 9, 5, and 0.
2
d. 0.28 · [​(5) ​ ​+ 13(5) + 42] = 0.28 · 132 = 36.96
The cost of fertilizer is $36.96.
e. 92+ 36.96 = 128.96
The total cost to fence and fertilize is $128.96.
2
7-4 factoring ​ax ​ ​+ bx + c
76.​x 2​ ​+ 2x - 24
b = 2 and c = -24; look for factors of -24 whose
sum is 2. The factor with the greater absolute value
is positive.
​Factors
of -24  
Sum​
__________________
-1 and 24 23 7
-2 and 12 10 7
-3 and 8 5 7
-4 and 6 2 3
The factors are -4 and 6.
(x - 4)(x + 6)
check it out!
1a. 6​x 2​ ​+ 11x + 3
(1x + 3)(6x + 1) = 6​x 2​ ​+ 19x + 3 7
(1x + 1)(6x + 3) = 6​x 2​ ​+ 9x + 3 7
(2x + 3)(3x + 1) = 6​x 2​ ​+ 11x + 3 3
The factors of 6​x ​2​+ 11x + 3 are (2x + 3) and
(3x + 1).
6​x ​2​+ 11x + 3 = (2x + 3)(3x + 1)
2
b. 3​x ​ ​- 2x - 8
(1x - 8)(3x + 1) = 3​x 2​ ​- 23x - 8 7
(1x - 4)(3x + 2) = 3​x 2​ ​-10x - 8 7
(1x + 4)(3x - 2) = 3​x 2​ ​+ 10x - 8 7
(1x + 8)(3x - 1) = 3​x 2​ ​+ 23x -8 7
(3x - 8)(1x + 1) = 3​x 2​ ​- 5x - 8 7
(3x - 4)(1x + 2) = 3​x 2​ ​+ 2x - 8 7
(3x + 4)(1x - 2) = 3​x 2​ ​- 2x - 8 3
The factors of 3​x ​2​- 2x - 8 are (3x + 4) and
(x - 2).
3​x ​2​- 2x - 8 = (3x + 4)(x - 2)
challenge and extend
4
2
77.​x 4​ ​+ 18​x 2​ ​+ 81
78.​y ​ ​- 5​y ​ ​- 24
​Factors
​Factors
of 81   
Sum​
of -24  
Sum​
________________
__________________
1 and 81 1 and -24 82
-23
3 and 27 30
2 and -12 -10
9 and 9 18
3 and -8 -5
(​x 2​ ​+ 9)(​x 2​ ​+ 9)
(​y 2​ ​+ 3)(​y 2​ ​- 8)
4
2
0.​(u + v) ​2​+ 2(u + v) - 3
8
79.​d ​ ​+ 22​d ​ ​+ 21
​Factors
​Factors
of 21   
Sum​
of -3   
Sum​
________________
_________________
1 and 21 -1 and 3 22
2
(​d ​2​+ 1)(​d ​2​+ 21)
(u + v - 1)(u + v + 3)
2
81.​(de) ​2​- (de) - 20
82.​(m - n) ​2​- 4(m - n) - 45
​Factors
​Factors
of
-20
  
Sum
of -45  
Sum
__________________
__________________
​1 and -20 ​1 and -45 -19
-44
2 and -10 -8
3 and -15 -12
4 and -5 -1
5 and -9 -4
(de + 4)(de - 5)
(m - n + 5)(m - n - 9)
2a. 6​x ​ ​+ 17x + 5
Factors of
​___________________________________
6 Factors
    
of 5   Outer + Inner​​
1 and 6 1 and 5
1(5) + 6(1) = 11 7
2 and 3 5 and 12(1) + 3(5) = 17 3
(2x + 5)(3x + 1)
b. 9​x 2​ ​- 15x + 4
​___________________________________
Factors
of 9 Factors
    
of 4 Outer + Inner
​1 and 9 -1 and -4 1(-4) + 9(-1) = -13 7
1 and 9 -4 and -1 1(-1) + 9(-4) = -37 7
1 and 9 -2 and -21(-2) + 9(-2) = -20 7
3 and 3 -1 and -4 3(-4) + 3(-1) = -15 3
(3x - 1)(3x - 4)
234 Holt McDougal Algebra 1
c. 3​x 2​ ​+ 13x + 12
​____________________________________
Factors
of 3 Factors
    
of 12 Outer + Inner​
1 and 3 1 and 12 1(12) + 3(1) = 15
1 and 3 12 and 1 1(1) + 3(12) = 37
1 and 3 2 and 6 1(6) + 3(2) = 12
1 and 3 6 and 2 1(2) + 3(6) = 20
1 and 3 3 and 4 1(4) + 3(3) = 13
(x + 3)(3x + 4)
Exercises
Guided Practice
7
7
7
7
3
3a. 6​x 2​ ​+ 7x - 3
​_
Factors
of 6 Factors
    
of -3 Outer + Inner​
___________________________________
1 and 6 1 and -3 1(-3) + 6(1) = 3 7
1 and 6 -1 and 3 1(3) + 6(-1) = -3 7
1 and 6 3 and -1 1(-1) + 6(3) = 17 7
1 and 6 -3 and 1 1(1) + 6(-3) = -17 7
2 and 3 1 and -3 2(-3) + 3(1) = -3 7
2 and 3 -1 and 3 2(3) + 3(-1) = 3 7
2 and 3 3 and -1 2(-1) + 3(3) = 7 3
(2x + 3)(3x - 1)
b. 4​n ​2​- n - 3
​_
Factors
of 4 Factors
    
of -3 Outer + Inner​
___________________________________
1 and 4 1 and -3 1(-3) + 4(1) = 1 7
1 and 4 -1 and 3 1(3) + 4(-1) = -1 3
(n - 1)(4n + 3)
4a. -6​x ​2​- 17x - 12
-1(6​x 2​ ​+ 17x + 12)
​Factors
of 6 Factors
    
of 12 Outer + Inner​
____________________________________
1 and 6 1 and 12 1(12) + 6(1) = 18 7
1 and 6 12 and 1 1(1) + 6(12) = 73 7
1 and 6 2 and 6 1(6) + 6(2) = 18 7
1 and 6 6 and 2 1(2) + 6(6) = 38 7
1 and 6 3 and 4 1(4) + 6(3) = 22 7
1 and 6 4 and 3 1(3) + 6(4) = 27 7
2 and 3 1 and 12 2(12) + 3(1) = 27 7
2 and 3 12 and 1 2(1) + 3(12) = 38 7
2 and 3 2 and 6 2(6) + 3(2) = 18 7
2 and 3 6 and 2 2(2) + 3(6) = 22 7
2 and 3
3 and 4 2(4) + 3(3) = 17 3
-1(2x + 3)(3x + 4)
b. -3​x ​2​- 17x - 10
-1(3​x 2​ ​+ 17x + 10)
​Factors
of 3 Factors
    
of 10 Outer + Inner​
____________________________________
1 and 3 1 and 10 1(10) + 3(1) = 12 7
1 and 3 10 and 1 1(1) + 3(10) = 31 7
1 and 3 2 and 5 1(5) + 3(2) = 11 7
1 and 3 5 and 2 1(2) + 3(5) = 17 3
-1(x + 5)(3x + 2)
1. 2​x 2​ ​+ 9x + 10
(1x + 10)(2x + 1) = 2​x 2​ ​+ 21x + 10 7
(1x + 5)(2x + 2)= 2​x 2​ ​+ 12x + 10 7
(1x + 2)(2x + 5) = 2​x 2​ ​+ 9x + 10 3
The factors of 2​x ​2​+ 9x + 10 are (x + 2) and
(2x + 5).
2​x ​2​+ 9x + 10 = (x + 2)(2x + 5)
2
2. 5​x ​ ​+ 31x + 6
(1x + 6)(5x + 1) = 5​x 2​ ​+ 31x + 6 3
The factors of 5​x ​2​+ 31x + 6 are (x + 6) and
(5x + 1).
5​x ​2​+ 31x + 6 = (x + 6)(5x + 1)
2
3. 5​x ​ ​+ 7x - 6
(1x - 6)(5x + 1) = 5​x 2​ ​- 29x - 6 7
(1x - 3)(5x + 2) = 5​x ​2​- 13x - 6 7
(1x - 2)(5x + 3) = 5​x ​2​- 7x - 6 7
(1x + 2)(5x - 3) = 5​x 2​ ​+ 7x - 6 3
The factors of 5​x ​2​+ 7x - 6 are (x + 2) and
(5x - 3).
5​x ​2​+ 7x - 6 = (x + 2)(5x - 3)
2
4. 6​x ​ ​+ 37x + 6
(1x + 6)(6x + 1) = 6​x 2​ ​+ 37x + 6 3
The factors of 6​x ​2​+ 37x + 6 are (x + 6) and
(6x + 1).
6​x ​2​+ 37x + 6 = (x + 6)(6x + 1)
2
5. 3​x ​ ​- 14x - 24
(1x - 24)(3x + 1) = 3​x 2​ ​- 71x - 24 7
(1x - 12)(3x + 2) = 3​x 2​ ​- 34x - 24 7
(1x - 8)(3x + 3) = 3​x 2​ ​- 21x - 24 7
(1x - 6)(3x + 4)= 3​x 2​ ​- 14x - 24 3
The factors of 3​x ​2​- 14x - 24 are (x - 6) and
(3x + 4).
3​x ​2​- 14x - 24 = (x - 6)(3x + 4)
2
6.​6x ​ ​+ x - 2
(1x - 2)(6x + 1) = 6​x 2​ ​- 11x - 2 7
(1x - 1)(6x + 2) = 6​x 2​ ​- 4x - 2 7
(1x + 1)(6x - 2) = 6​x 2​ ​+ 4x -2 7
(1x + 2)(6x - 1) = 6​x ​2​+ 11x - 2 7
(2x - 2)(3x + 1) = 6​x 2​ ​- 4x -2 7
(2x - 1)(3x + 2) = 6​x 2​ ​+ x -2 3
The factors of 6​x ​2​+ x - 2 are (2x - 1) and
(3x + 2).
6​x ​2​+ x - 2 = (2x - 1)(3x + 2).
2
think and discuss
1. The signs of the numbers are all positive.
2.
Factoring ax 2 + bx + c
c>0
b>0
b<0
3x 2 + 10x + 8 = (3x + 4)(x + 2)
3x 2 - 10x + 8 = (3x - 4)(x - 2)
c<0
b<0
2
3x - 10x - 8 = (3x + 2)(x - 4)
7. 5​x ​ ​+ 11x + 2
​___________________________________
Factors
of 5 Factors
    
of 2 Outer + Inner​
1 and 5 1 and 2 1(2) + 5(1) = 7 7
1 and 5 2 and 1 1(1) + 5(2) = 11 3
(x + 2)(5x + 1)
8. 2​x 2​ ​+ 11x + 5
​___________________________________
Factors
of 2 Factors
    
of 5 Outer + Inner​
1 and 2 1 and 5 1(5) + 2(1) = 7 7
1 and 2 5 and 1
1(1) + 2(5) = 11 3
(x + 5)(2x + 1)
b>0
2
3x + 10x - 8 = (3x - 2)(x + 4)
235 Holt McDougal Algebra 1
9. 4​x 2​ ​- 9x + 5
​ __________________________________
Factors
of 4 Factors
    
of 5 Outer + Inner​
_
1 and 4 -1 and -5 1(-5) + 4(-1) = -9 3
(x - 1)(4x - 5)
2
10. 2​y ​ ​- 11y + 14
​ __________________________________
Factors
of 2 Factors
    
of 14 Outer + Inner​
_
1 and 2 -1 and -141(-14) + 2(-1) = -167
1 and 2 -14 and -1 1(-1) + 2(-14) = -297
1 and 2 -2 and -7 1(-7) + 2(-2) = -113
(2y - 7)(y - 2)
2
11. 5​x ​ ​+ 9x + 4
​___________________________________
Factors
of 5 Factors
    
of 4 Outer + Inner​
1 and 5 1 and 4 1(4) + 5(1) = 9 3
(x + 1)(5x + 4)
2
12. 3​x ​ ​+ 7x + 2
​___________________________________
Factors
of 3 Factors
    
of 2 Outer + Inner​
1 and 3 1 and 2 1(2) + 3(1) = 5 7
1 and 3 2 and 1 1(1) + 3(2) = 7 3
(x + 2)(3x + 1)
13. 4​a 2​ ​+ 8a - 5
​_
Factors
of 4 Factors
    
of -5 Outer + Inner
___________________________________
​1 and 4 1 and -5 1(-5) + 4(1) = -1 7
1 and 4 -1 and 5 1(5) + 4(-1) = 1 7
1 and 4 5 and -1 1(-1) + 4(5) = 19 7
1 and 4 -5 and 1 1(1) + 4(-5) = -19 7
2 and 2 1 and -5 2(-5) + 2(1) = - 8 7
2 and 2 -1 and 5 2(5) + 2(-1) = 8 3
(2a - 1)(2a + 5)
14. 15​x ​2​+ 4x - 3
​____________________________________
Factors
of 15 Factors
    
of -3 Outer + Inner
​1 and 15 1 and -3 1(-3) + 15(1) = 12 7
1 and 15 -1 and 3 1(3) + 15(-1) = -127
1 and 15 3 and -1 1(-1) + 15(3) = 44 7
1 and 15 -3 and 1 1(1) + 15(-3) = -447
3 and 5 1 and -3 3(-3) + 5(1) = -4 7
3 and 5 -1 and 3 3(3) + 5(-1) = 4 3
(3x - 1)(5x + 3)
15. 2​x 2​ ​+ x - 6
​_
Factors
of 2 Factors
    
of -6 Outer + Inner​
___________________________________
1 and 2 1 and -6 1(-6) + 2(1) = -4 7
1 and 2 -1 and 6 1(6) + 2(-1) = 4 7
1 and 2
2 and -3 1(-3) + 2(2) = 1 3
(x + 2)(2x - 3)
16. 6​n ​2​- 11n - 10
​____________________________________
Factors
of 6 Factors
    
of -10 Outer + Inner​
1 and 6 1 and -10 1(-10) + 6(1) = -47
1 and 6 -1 and 10 1(10) + 6(-1) = 47
1 and 6 2 and -5 1(-5) + 6(2) = 7 7
1 and 6 -2 and 5 1(5) + 6(-2) = -77
1 and 6 5 and -2 1(-2) + 6(5) = 287
1 and 6 -5 and 2 1(2) + 6(-5) = -287
1 and 6 10 and -1 1(-1) + 6(10) = 597
1 and 6 -10 and 1 1(1) + 6(-10) = -597
2 and 3 2 and -5 2(-5) + 3(2) = -47
2 and 3
-2 and 5 2(5) + 3(-2) = 47
2 and 3 5 and -2 2(-2) + 3(5) = 117
2 and 3 -5 and 2 2(2) + 3(-5) = -113
(2n - 5)(3n + 2)
17. 10​x 2​ ​- 9x - 1
​_____________________________________
Factors
of 10     
Factors of -1 Outer + Inner​
1 and 10 1 and -1 1(-1) + 10(1) = 9 7
1 and 10 -1 and 1 1(1) + 10(-1) = -93
(x - 1)(10x + 1)
18. 7​x 2​ ​- 3x - 10
​_____________________________________
Factors
of 7      
Factors of -10 Outer + Inner
​1 and 7 1 and -10 1(-10) + 7(1) = -33
(x + 1)(7x - 10)
2
19. -2​x ​ ​+ 5x + 12
-1(2​x 2​ ​- 5x - 12)​
Factors
of 2 Factors
    
of -12 Outer + Inner​
____________________________________
1 and 2 1 and -12 1(-12) + 2(1) = -107
1 and 2 -1 and 12 1(12) + 2(-1) = 10 7
1 and 2 2 and -6 1(-6) + 2(2) = -2 7
1 and 2 -2 and 6 1(6) + 2(-2) = 2 7
1 and 2 3 and -4 1(-4) + 2(3) = 2 7
1 and 2 -3 and 4 1(4) + 2(-3) = -2 7
1 and 2 4 and -3 1(-3) + 2(4) = 5 7
1 and 2 -4 and 3 1(3) + 2(-4) = -53
= -1(2x + 3)(x - 4)
20. -4​n ​2​- 16n + 9
-1(4​n ​2​+ 16n - 9)
​Factors
of 4 Factors
    
of -9 Outer + Inner​
____________________________________
1 and 4 1 and -9 1(-9) + 4(1) = -5 7
1 and 4 -1 and 9 1(9) + 4(-1) = 5 7
1 and 4 3 and -3 1(-3) + 4(3) = 9 7
1 and 4
-3 and 3 1(3) + 4(-3) = -9 7
1 and 4 9 and -1 1(-1) + 4(9) = 35 7
1 and 4 -9 and 1 1(1) + 4(-9) = -35 7
2 and 2 1 and -9 2(-9) + 2(1) = -16 7
2 and 2 -1 and 9 2(9) + 2(-1) = 16 3
-1(2n - 1)(2n + 9)
21. -5​x 2​ ​+ 7x + 6
-1(5​x ​2​- 7x - 6)
​Factors
of 5 Factors
    
of -6 Outer + Inner​
____________________________________
1 and 5 1 and -6 1(-6) + 5(1) = -1 7
1 and 5 -1 and 6 1(6) + 5(-1) = 1 7
1 and 5 2 and -3 1(-3) + 5(2) = 7 7
1 and 5 -2 and 3 1(3) + 5(-2) = -7 3
-1(x - 2)(5x + 3)
22. -6​x 2​ ​+ 13x - 2
-1(6​x ​2​- 13x + 2)
​___________________________________
Factors
of 6 Factors
    
of 2 Outer + Inner
​1 and 6 -1 and -2 1(-2) + 6(-1) = -8 7
1 and 6 -2 and -1 1(-1) + 6(-2) = -13 3
-1(x - 2)(6x - 1)
23. -4​x 2​ ​- 8x + 5
-1(4​x ​2​+ 8x - 5)
​Factors
of 4 Factors
    
of -5 Outer + Inner​
____________________________________
1 and 4 1 and -5 1(-5) + 4(1) = -1 7
1 and 4 -1 and 5 1(5) + 4(-1) = 1 7
1 and 4 5 and -1 1(-1) + 4(5) = 19 7
1 and 4 -5 and 1 1(1) + 4(-5) = -19 7
2 and 2 1 and -5 2(-5) + 2(1) = -8 7
2 and 2 -1 and 5 2(5) + 2(-1) = 8 3
-1(2x - 1)(2x + 5)
236 Holt McDougal Algebra 1
24. -5​x 2​ ​+ x + 18
-1(5​x 2​ ​- x - 18)
​Factors
of 5 Factors
    
of -18 Outer + Inner​
____________________________________
1 and 5 1 and -18 1(-18) + 5(1) = -137
1 and 5 -1 and 18 1(18) + 5(-1) = 13 7
1 and 5 2 and -9 1(-9) + 5(2) = 1 7
1 and 5 -2 and 9 1(9) + 5(-2) = -1 3
-1(x - 2)(5x + 9)
practice and problem solving
25. 9​x 2​ ​+ 9x + 2
(1x + 2)(9x + 1) = 9​x 2​ ​+ 19x + 2 7
(1x + 1)(9x + 2) = 9​x 2​ ​+ 11x + 2 7
(3x + 2)(3x + 1) = 9​x 2​ ​+ 9x + 2 3
The factors of 9​x ​2​+ 9x + 2 are (3x + 2) and
(3x + 1).
9​x ​2​+ 9x + 2 = (3x + 2)(3x + 1)
2
26. 2​x ​ ​+ 7x + 5
(1x + 5)(2x + 1) = 2​x 2​ ​+ 11x + 5 7
(1x + 1)(2x + 5) = 2​x 2​ ​+ 7x + 5 3
The factors of 2​x ​2​+ 7x + 5 are (x + 1) and
(2x + 5).
2​x ​2​+ 7x + 5 = (x + 1)(2x + 5)
2
27. 3​n ​ ​+ 8n + 4
(1n + 4)(3n + 1) = 3​n 2​ ​+ 13n + 4 7
(1n + 2)(3n + 2) = 3​n ​2​+ 8n + 4 3
The factors of 3​n ​2​+ 8n + 4 are (n + 2) and
(3n + 2).
3​n ​2​+ 8n + 4 = (n + 2)(3n + 2)
2
28. 10​d ​ ​+ 17d + 7
(1d + 7)(10d + 1) = 10​d 2​ ​+ 71d + 7 7
(1d + 1)(10d + 7) = 10​d 2​ ​+ 17d + 7 3
The factors of 10​d ​2​+ 17d + 7 are (d + 1) and
(10d + 7).
10​d ​2​+ 17d + 7 = (d + 1)(10d + 7)
2
29. 4​c ​ ​- 17c + 15
(1c - 15)(4c - 1) = 4​c 2​ ​- 61c + 15 7
(1c - 5)(4c - 3)= 4​c ​2​- 23c + 15 7
(1c - 3)(4c - 5)= 4​c ​2​- 17c + 15 3
The factors of 4​c ​2​- 17c + 15 are (c - 3) and
(4c - 5).
4​c ​2​- 17c + 15 = (c - 3)(4c - 5)
2
30. 6​x ​ ​+ 14x + 4
(1x + 4)(6x + 1) = 6​x 2​ ​+ 25x + 4 7
(1x + 2)(6x + 2) = 6​x 2​ ​+ 14x + 4 3
The factors of 6​x ​2​+ 14x + 4 are (x + 2) and
(6x + 2).
6​x ​2​+ 14x + 4 = 2(3x + 1)(x + 2)
2
31. 8​x ​ ​+ 22x + 5
(1x + 5)(8x + 1) = 8​x 2​ ​+ 41x + 5 7
(1x + 1)(8x + 5) = 8​x 2​ ​+ 13x + 5 7
(2x + 5)(4x + 1) = 8​x 2​ ​+ 22x + 5 3
The factors of 8​x ​2​+ 22x + 5 are (2x + 5) and
(4x + 1).
8​x ​2​+ 22x + 5 = (2x + 5)(4x + 1)
32. 6​x ​2​- 13x + 6
(1x - 6)(6x - 1) = 6​x 2​ ​- 37x + 6 7
(1x - 3)(6x - 2) = 6​x 2​ ​- 20x + 6 7
(1x - 2)(6x - 3) = 6​x 2​ ​- 15x + 6 7
(1x - 1)(6x - 6) = 6​x 2​ ​- 12x + 6 7
(2x - 6)(3x - 1) = 6​x 2​ ​- 20x + 6 7
(2x - 3)(3x - 2) = 6​x 2​ ​- 13x + 6 3
The factors of 6​x ​2​- 13x + 6 are (2x - 3) and
(3x - 2).
6​x ​2​- 13x + 6 = (2x - 3)(3x - 2)
2
33. 5​x ​ ​+ 9x - 18
(1x + 18)(5x - 1) = 5​x 2​ ​+ 89x - 18 7
(1x + 9)(5x - 2)= 5​x 2​ ​+ 43x - 18 7
(1x + 6)(5x - 3)= 5​x 2​ ​+ 27x - 18 7
(1x + 3)(5x - 6)= 5​x  ​2​+ 9x - 18 3
The factors of 5​x ​2​+ 9x - 18 are (x + 3) and
(5x - 6).
5​x ​2​+ 9x - 18 = (x + 3)(5x - 6)
2
34. 6​x ​ ​+ 23x + 7
​___________________________________
Factors
of 6 Factors
    
of 7 Outer + Inner​
1 and 6 1 and 7 1(7) + 6(1) = 13
1 and 6 7 and 1 1(1) + 6(7) = 43
2 and 3 1 and 7 2(7) + 3(1) = 17
2 and 3 7 and 1 2(1) + 3(7) = 23
(2x + 7)(3x + 1)
7
7
7
3
35. 10​n 2​ ​- 17n + 7
​___________________________________
Factors
of 10 Factors
    
of 7 Outer + Inner​
1 and 10 -1 and -7 1(-7) + 10(-1) = -173
(n - 1)(10n - 7)
2
36. 3​x ​ ​+ 11x + 6
​___________________________________
Factors
of 3 Factors
    
of 6 Outer + Inner​
1 and 3 1 and 6 1(6) + 3(1) = 9
1 and 3 6 and 1 1(1) + 3(6) = 19
1 and 3 2 and 3 1(3) + 3(2) = 9
1 and 3 3 and 2 1(2) + 3(3) = 11
(x + 3)(3x + 2)
7
7
7
3
37. 7​x 2​ ​+ 15x + 2
​___________________________________
Factors
of 7 Factors
    
of 2 Outer + Inner​
1 and 7 1 and 2 1(2) + 7(1) = 9 7
1 and 7 2 and 1 1(1) + 7(2) = 15 3
(x + 2)(7x + 1)
38. 3​n 2​ ​+ 4n + 1
​___________________________________
Factors
of 3 Factors
    
of 1 Outer + Inner​
1 and 3 1 and 1 1(1) + 3(1) = 4 3
(n + 1)(3n + 1)
2
39. 3​x ​ ​- 19x + 20
​___________________________________
Factors
of 3 Factors
    
of 20 Outer + Inner​
1 and 3 -1 and -20 1(-20) + 3(-1) = -237
1 and 3 -20 and -1 1(-1) + 3(-20) = -617
1 and 3 -2 and -101(-10) + 3(-2) = -167
1 and 3 -10 and -2 1(-2) + 3(-10) = -327
1 and 3 -4 and -5 1(-5) + 3(-4) = -177
1 and 3 -5 and -4 1(-4) + 3(-5) = -193
(x - 5)(3x - 4)
237 Holt McDougal Algebra 1
40. 6​x 2​ ​+ 11x + 4
​ __________________________________
Factors
of 6 Factors
    
of 4 Outer + Inner​
_
1 and 6 1 and 4 1(4) + 6(1) = 10
1 and 6 4 and 1 1(1) + 6(4) = 25
1 and 6 2 and 2 1(2) + 6(2) = 14
2 and 3 1 and 4 2(4) + 3(1) = 11
(2x + 1)(3x + 4)
7
7
7
3
41. 4​x 2​ ​- 31x + 21
​ __________________________________
Factors
of 4 Factors
    
of 21 Outer + Inner​
_
1 and 4 -1 and -211(-21) + 4(-1) = -257
1 and 4 -21 and -1 1(-1) + 4(-21) = -857
1 and 4 -3 and -7 1(-7) + 4(-3) = -197
1 and 4 -7 and -3 1(-3) + 4(-7) = -313
(x - 7)(4x - 3)
42. 10​x ​2​+ 31x + 15
​_____________________________________
Factors
of 10 Factors
    
of 15 Outer + Inner
​1 and 10 1 and 15 1(15) + 10(1) = 25 7
1 and 10
15 and 1 1(1) + 10(15) = 1517
1 and 10
3 and 5 1(5) + 10(3) = 35 7
1 and 10 5 and 3 1(3) + 10(5) = 53 7
2 and 5
1 and 15 2(15) + 5(1) = 35 7
2 and 5 15 and 1 2(1) + 5(15) = 77 7
2 and 5 3 and 5 2(5) + 5(3) = 25 7
2 and 5
5 and 3 2(3) + 5(5) = 31 3
(2x + 5)(5x + 3)
43. 12​y 2​ ​+ 17y - 5
​____________________________________
Factors
of 12 Factors
    
of -5 Outer + Inner​
1 and 12 1 and -5 1(-5) + 12(1) = 7 7
1 and 12 -1 and 5 1(5) + 12(-1) = -7 7
1 and 12 5 and -1 1(-1) + 12(5) = 59 7
1 and 12 -5 and 1 1(1) + 12(-5) = -597
2 and 6 1 and -5 2(-5) + 6(1) = -4 7
2 and 6 -1 and 5 2(5) + 6(-1) = 4 7
2 and 6 5 and -1 2(-1) + 6(5) = 28 7
2 and 6 -5 and 1 2(1) + 6(-5) = -287
3 and 4
1 and -5 3(-5) + 4(1) = -117
3 and 4 -1 and 5 3(5) + 4(-1) = 11 7
3 and 4 5 and -1 3(-1) + 4(5) = 17 3
(3y + 5)(4y - 1)
44. 3​x 2​ ​+ 10x - 8
​_
Factors
of 3 Factors
    
of -8 Outer + Inner​
___________________________________
1 and 3 1 and -8 1(-8) + 3(1) = -57
1 and 3 -1 and 8 1(8) + 3(-1) = 5 7
1 and 3 2 and -4 1(-4) + 3(2) = 2 7
1 and 3 -2 and 4 1(4) + 3(-2) = -2 7
1 and 3 4 and -2 1(-2) + 3(4) = 10 3
(x + 4)(3x - 2)
45. 4​x 2​ ​+ 4x - 3
​_
Factors
of 4 Factors
    
of -3 Outer + Inner
___________________________________
​1 and 4 1 and -3 1(-3) + 4(1) = 1 7
1 and 4 -1 and 3 1(3) + 4(-1) = -1 7
1 and 4 3 and -1 1(-1) + 4(3) = 11 7
1 and 4 -3 and 1 1(1) + 4(-3) = -11 7
2 and 2 1 and -3 2(-3) + 2(1) = -4 7
2 and 2 -1 and 3 2(3) + 2(-1) = 4 3
(2x - 1)(2x + 3)
46. 2​n 2​ ​- 7n - 4
​_
Factors
of 2 Factors
    
of -4 Outer + Inner​
___________________________________
1 and 2 1 and -4 1(-4) + 2(1) = -2 7
1 and 2 -1 and 4 1(4) + 2(-1) = 2 7
1 and 2 2 and -2 1(-2) + 2(2) = 2 7
1 and 2 -2 and 2 1(2) + 2(-2) = -2 7
1 and 2 4 and -1 1(-1) + 2(4) = 7 7
1 and 2 -4 and 1 1(1) + 2(-4) = -7 3
(n - 4)(2n + 1)
47. 3​x 2​ ​- 4x - 15
​_
Factors
of 3 Factors
    
of -15 Outer + Inner​
___________________________________
1 and 3 1 and -15 1(-15) + 3(1) = -127
1 and 3 -1 and 15 1(15) + 3(-1) = 12 7
1 and 3 3 and -5 1(-5) + 3(3) = 4 7
1 and 3 -3 and 5 1(5) + 3(-3) = -4 3
(x - 3)(3x + 5)
48. 3​n ​2​- n - 4
​_
Factors
of 3 Factors
    
of -4 Outer + Inner​
___________________________________
1 and 3 1 and -4 1(-4) + 3(1) = -1 3
(n + 1)(3n - 4)
2
49. -4​x ​ ​- 4x + 15
-1(4​x ​2​+ 4x - 15)
​_
Factors
of 4 Factors
    
of -15 Outer + Inner​
___________________________________
1 and 4 1 and -15 1(-15) + 4(1) = -117
1 and 4 -1 and 15 1(15) + 4(-1) = 11 7
1 and 4 3 and -5 1(-5) + 4(3) = 7 7
1 and 4 -3 and 5 1(5) + 4(-3) = -7 7
1 and 4 5 and -3 1(-3) + 4(5) = 17 7
1 and 4 -5 and 3 1(3) + 4(-5) = -177
1 and 4 15 and -1 1(-1) + 4(15) = 59 7
1 and 4 -15 and 1 1(1) + 4(-15) = -597
2 and 2 1 and -152(-15) + 2(1) = -287
2 and 2 -1 and 15 2(15) + 2(-1) = 28 7
2 and 2 3 and -5 2(-5) + 2(3) = -4 7
2 and 2 -3 and 5 2(5) + 2(-3) = 4 3
-1(2x - 3)(2x + 5)
50. -3​x ​2​+ 16x - 16
-1(3​x 2​ ​- 16x + 16)
​___________________________________
Factors
of 3 Factors
    
of 16 Outer + Inner​
1 and 3 -1 and -161(-16) + 3(-1) = -197
1 and 3 -16 and -1 1(-1) + 3(-16) = -497
1 and 3 -2 and -8 1(-8) + 3(-2) = -147
1 and 3 -8 and -2 1(-2) + 2(-8) = -187
1 and 3 -4 and -4 1(-4) + 3(-4) = -163
-1(x - 4)(3x - 4)
51. -3​x 2​ ​- x + 2
-1(3​x 2​ ​+ x - 2)
​_
Factors
of 3 Factors
    
of -2 Outer + Inner
___________________________________
​1 and 3 1 and -2 1(-2) + 3(1) = 1 3
-1(x + 1)(3x - 2)
52. 12​x ​2​+ 24x + 3x + 6 53. 2​x ​2 ​- 4x - x + 2
(12​x 2​ ​+ 24x) + (3x + 6)
(2​x ​2​- 4x) - (x - 2)
12x(x + 2) + 3(x + 2)
2x(x - 2) - (x - 2)
12​x ​2​+ 27 + 6;
(2x - 1)(x - 2)
3(4x + 1)(x + 2)
54. 5​x ​2​+ 35x - 4x - 28
(5​x 2​ ​+ 35x) - (4x + 28)
5x(x + 7) - 4(x + 7)
(5x - 4)(x + 7)
238 Holt McDougal Algebra 1
55. 9​n 2​ ​+ 17n + 8
(1n + 8)(9n + 1) = 9​n 2​ ​+ 73n + 8 7
(1n + 4)(9n + 2) = 9​n 2​ ​+ 38n + 8 7
(1n + 2)(9n + 4) = 9​n 2​ ​+ 22n + 8 7
(1n + 1)(9n + 8) = 9​n 2​ ​+ 17n + 8 3
The factors of 9​n ​2​+ 17n + 8 are (n + 1) and
(9n + 8).
9​n ​2​+ 17n + 8 = (n + 1)(9n + 8)
2
56. 2​x ​ ​- 7x - 4
(1x - 4)(2x + 1) = 2​x 2​ ​- 7x - 4 3
The factors of 2​x ​2​- 7x - 4 are (x - 4) and
(2x + 1).
2​x ​2​- 7x - 4 = (x - 4)(2x + 1)
2
57. 4​x ​ ​- 12x + 5
(1x - 5)(4x - 1) = 4​x 2​ ​- 21x + 5 7
(1x - 1)(4x - 5) = 4​x 2​ ​- 9x + 5 7
(2x - 5)(2x - 1) = 4​x 2​ ​- 12x + 5 3
The factors of 4​x ​2​- 12x + 5 are (2x - 5) and
(2x - 1).
4​x ​2​- 12x + 5 = (2x - 5)(2x - 1)
2
58. 5​x ​ ​- 4x + 12
(1x - 12)(5x - 1) = 5​x 2​ ​- 61x + 12 7
(1x - 6)(5x - 2)= 5​x ​2​- 32x + 12 7
(1x - 4)(5x - 3)= 5​x ​2​- 23x + 12 7
(1x - 3)(5x - 4)= 5​x 2​ ​- 19x + 12 7
(1x - 2)(5x - 6)= 5​x ​2​- 16x + 12 7
(1x - 1)(5x - 12) = 5​x 2​ ​- 17x + 12 7
(5​x ​2​- 4x + 12) cannot be factored.
2
59. 3​x ​ ​+ 14x + 16
(1x + 16)(3x + 1) = 3​x 2​ ​+ 49x + 16 7
(1x + 8)(3x + 2)= 3​x ​2​+ 26x + 16 7
(1x + 4)(3x + 4)= 3​x 2​ ​+ 16x + 16 7
(1x + 2)(3x + 8)= 3​x ​2​+ 14x + 16 3
The factors of 3​x ​2​+ 14x + 16 are (x + 2) and
(3x + 8).
3​x ​2​+ 14x + 16 = (x + 2)(3x + 8)
2
60. -3​x ​ ​- 11x + 4
-1(3​x 2​ ​+ 11x - 4)
(1x - 4)(3x + 1) = 3​x 2​ ​- 11x - 4 7
(1x + 4)(3x - 1) = 3​x 2​ ​+ 11x - 4 3
The factors of 3​x ​2​+ 11x - 4 are (x + 4) and
(3x - 1).
-3​x 2​ ​- 11x + 4 = -1(x + 4)(3x - 1)
61. 6​x 2​ ​- x - 12
(1x - 12)(6x + 1) = 6​x 2​ ​- 71x - 12 7
(1x + 12)(6x - 1) = 6​x 2​ ​+ 71x - 12 7
(1x - 6)(6x + 2)= 6​x ​2​- 34x - 12 7
(1x + 6)(6x - 2)= 6​x 2​ ​+ 34x - 12 7
(1x - 4)(6x + 3)= 6​x 2​ ​- 21x - 12 7
(1x + 4)(6x - 3)= 6​x 2​ ​+ 21x - 12 7
(1x - 3)(6x + 4)= 6​x 2​ ​- 14x - 12 7
(1x + 3)(6x - 4)= 6​x 2​ ​+ 14x - 12 7
(1x - 2)(6x + 6)= 6​x 2​ ​- 6x - 12 7
(1x + 2)(6x - 6)= 6​x 2​ ​+ 6x - 12 7
(1x - 1)(6x + 12) = 6​x 2​ ​+ 6x - 12 7
(1x + 1)(6x - 12) = 6​x 2​ ​- 6x - 12 7
(2x - 12)(3x + 1) = 6​x 2​ ​- 34x - 12 7
(2x + 12)(3x - 1) = 6​x 2​ ​+ 34x - 12 7
(2x - 6)(3x + 2)= 6​x 2​ ​- 14x - 12 7
(2x + 6)(3x - 2)= 6​x 2​ ​+ 14x - 12 7
(2x - 4)(3x + 3)= 6​x 2​ ​- 6x - 12 7
(2x + 4)(3x - 3)= 6​x 2​ ​+ 6x - 12 7
(2x - 3)(3x + 4)= 6​x 2​ ​- x - 12 3
The factors of 6​x ​2​- x - 12 are (2x - 3) and
(3x + 4).
6​x ​2​- x - 12 = (2x - 3)(3x + 4)
2
62. 10​a ​ ​+ 11a + 3
(1a + 3)(10a + 1) = 10​a 2​ ​+ 31a + 3 7
(1a + 1)(10a + 3) = 10​a 2​ ​+ 13a + 3 7
(2a + 3)(5a + 1)= 10​a 2​ ​+ 17a + 3 7
(2a + 1)(5a + 3)= 10​a 2​ ​+ 11a + 3 3
The factors of 10​a ​2​+ 11a + 3 are (2a + 1) and
(5a + 3).
10​a ​2​+ 11a + 3 = (2a + 1)(5a + 3)
2
63. 4​x ​ ​- 12x + 9
(1x - 9)(4x - 1) = 4​x 2​ ​- 37x + 9 7
(1x - 3)(4x - 3) = 4​x 2​ ​- 15x + 9 7
(1x - 1)(4x - 9) = 4​x 2​ ​- 13x + 9 7
(2x - 9)(2x - 2) = 4​x 2​ ​- 22x + 9 7
(2x - 3)(2x - 3) = 4​x 2​ ​- 12x + 9 3
The factors of 4​x ​2​- 12x + 9 are (2x - 3) and
(2x - 3).
4​x ​2​- 12x + 9 = (2x - 3)(2x - 3)
2
64. 6​x ​ ​+ 11x + 5
(1x + 6)(6x + 1) = 6​x 2​ ​+ 37x + 5
(1x + 1)(6x + 5) = 6​x 2​ ​+ 11x + 5
The factors of 6​x ​2​+ 11x + 5 are (x + 1) and
(6x + 5).
The length of the rectangle is (6x + 5) cm.
65. 6​x ​2​+ 13x + 6
a = 6 and c = 6; Outer + Inner = 13
​___________________________________
Factors
of 6 Factors
    
of 6 Outer + Inner
​1 and 6 1 and 6 1(6) + 6(1) = 12 7
1 and 6 6 and 1 1(1) + 6(6) = 37 7
1 and 6 2 and 3 1(3) + 6(2) = 15 7
1 and 6 3 and 2 1(2) + 6(3) = 20 7
2 and 3 1 and 6 2(6) + 3(1) = 15 7
2 and 3 6 and 1 2(1) + 3(6) = 20 7
2 and 3 2 and 3 2(3) + 3(2) = 12 7
2 and 3 3 and 2 2(2) + 3(3) = 13 3
(2x + 3)(3x + 2)
239 Holt McDougal Algebra 1
67. 4​x 2​ ​+ 9x + 2
66. 8​x 2​ ​+ 18x - 5
2
8​x ​ ​+ 20x - 2x - 5
4​x 2​ ​+ 8x + x + 2
2
(8​x ​ ​+ 20x) + (-2x - 5)
(4​x ​2​+ 8x) + (x + 2)
4x(2x + 5) - 1(2x + 5)
4x(x + 2) + 1(x + 2)
(4x - 1)(2x + 5)
(4x + 1)(x + 2)
test prep
68. 2​w 2​ ​+ 7w + 6
(1w + 6)(2w + 1) = 2​w 2​ ​+ 13w + 6
(1w + 3)(2w + 2) = 2​w 2​ ​+ 8w + 6
(1w + 2)(2w + 3) = 2​w 2​ ​+ 7w + 6
The length of Rebecca’s old garden is (2w) yd, and
the width is (w)yd.
The length of Rebecca’s new garden is (2w + 3) yd,
and the width is (w + 2) yd.
Length increased by 3 yd, and width increased by
2 yd.
9a. v = 20, h = 6
6
-16​t ​2​+ vt + h = -16​t 2​ ​+ 20t + 6
2
b. -16​t ​ ​+ 20t + 6
-2(8​t 2​ ​- 10t - 3)
(1t + 1)(8t - 3) = 8​t 2​ ​+ 5t - 3 7
(1t - 1)(8t + 3) = 8​t 2​ ​- 5t - 3 7
(1t + 3)(8t - 1) = 8​t 2​ ​+ 23t - 3 7
(1t - 3)(8t + 1) = 8​t 2​ ​- 23t - 3 7
(2t + 1)(4t - 3) = 8​t 2​ ​- 2t - 3 7
(2t - 1)(4t + 3) = 8​t ​2​+ 2t - 3 7
(2t + 3)(4t - 1) = 8​t 2​ ​+ 10t - 3 7
(2t - 3)(4t + 1) = 8​t 2​ ​- 10t - 3 3
-2(4t + 1)(2t - 3)
c.-1(2t - 3)(8t + 2) = -1(2(1) - 3)(8(1) + 2)
= -1(-1)(10) = 10
The height of the football after 1 second is 10 ft.
70. Possible answer: The student tried factors of 12
instead of factors of 2 · 12.
2
1a.
7
2​t ​ ​= 10t - 8
2
2​t ​ ​- 10t + 8 = 0
2
b. 2​t ​ ​- 10t + 8
2(​t 2​ ​- 5t + 4)
(t - 4)(t - 1) = t​ 2​ ​- 5t + 4 3
The factors of t​ ​2​- 5t + 4 are (t - 4) and (t - 1).
2​t ​2​- 10t + 8 = 2(t - 4)(t - 1)
c. The boats are the same distance from the start
2
point w
hen 2​t ​ ​- 10t + 8 = 0.
From factorization, 2(t - 4)(t - 1) = 0, so
(t - 4) = 0 or (t - 1) = 0.
Therefore the boats are the same distance from the
start point when t = 1 and t = 4.
72. D;
(x - 5)(6x + 1)
6​x 2​ ​+ x - 30x - 5
6​x ​2​- 29x - 5
73. B;
(x - 5)(6x - 1)
6​x ​2​- x - 30x + 5
6​x ​2​- 31x + 5
75. C;
74. A;
(x + 5)(6x - 1)
(x + 5)(6x - 1)
6​x ​2​- x + 30x - 5
6​x ​2​- x + 30x - 5
6​x ​2​+ 29x - 5
6​x ​2​+ 29x - 5
76a. Both signs are positive, or both signs are negative.
b. One sign is positive, and the other is negative.
77. B;
3​x 2​ ​+ bx - 8
​_
Factors
of 3 Factors
    
of -8 Outer + Inner
___________________________________
​1 and 3 1 and -8 1(-8) + 3(1) = -5
1 and 3 -1 and 8 1(8) + 3(-1) = 5
1 and 3 2 and -4 1(-4) + 3(2) = 2
1 and 3 -2 and 4 1(4) + 3(-2) = -2
1 and 3 4 and -2 1(-2) + 3(4) = 10
1 and 3 -4 and 2 1(2) + 3(-4) = -10
1 and 3 8 and -1 1(-1) + 3(8) = 23
1 and 3 -8 and 1 1(1) + 3(-8) = -23
Possible values of b are -23, -10, -5, -2, 2, 5,
10, and 23.
78. H;
5​x 2​ ​+ 15x + 4x + 12
(5​x 2​ ​+ 15x) + (4x + 12)
5x(x + 3) + 4(x + 3)
(5x + 4)(x + 3)
79. A;
24​x ​2​- 49x + 2
(1x - 2)(24x -1) = 24​x 2​ ​- 49x + 2 3
The factors of 24​x ​2​- 49x + 2 are (x - 2) and
(24x - 1).
80. G;
c = -15; 2​x ​2​+ x - 15
(1x + 1)(2x - 15) = 2​x 2​ ​- 13x - 15 7
(1x - 1)(2x + 15) = 2​x 2​ ​+ 13x - 15 7
(1x + 3)(2x - 5)= 2​x 2​ ​+ x - 15 3
The factors of 2​x ​2​+ x - 15 are (x + 3) and
(2x - 5).
c = -9; 2​x ​2​+ x - 9
(1x + 1)(2x - 9) = 2​x 2​ ​- 7x - 9 7
(1x - 1)(2x + 9) = 2​x ​2​+ 7x - 9 7
(1x + 3)(2x - 3) = 2​x ​2​+ 3x - 9 7
(1x - 3)(2x + 3) = 2​x ​2​- 3x - 9 7
(1x + 9)(2x - 1) = 2​x 2​ ​+ 17x - 9 7
(1x - 9)(2x + 1) = 2​x 2​ ​- 17x - 9 7
(2​x ​2​+ x - 9) cannot be factored.
c = -6; 2​x 2​ ​+ x - 6
(1x + 1)(2x - 6) = 2​x 2​ ​- 4x - 6 7
(1x - 1)(2x + 6) = 2​x 2​ ​+ 4x - 6 7
(1x + 2)(2x - 3) = 2​x 2​ ​+ x -6 3
The factors of 2​x ​2​+ x - 6 are (x + 2) and (2x - 3).
c = -1; 2​x ​2​+ x - 1
(1x + 1)(2x - 1) = 2​x 2​ ​+ x -1 3
The factors of 2​x ​2​+ x - 1 are (x + 1) and (2x - 1).
challenge and extend
81. 1 + 4x + 4​x 2​ ​
​___________________________________
Factors
of 4 Factors
    
of 1 Outer + Inner
​1 and 4 1 and 1 1(1) + 4(1) = 5 7
2 and 2 1 and 1 1(2) + 2(1) = 4 3
(2x + 1)(2x + 1)
82. 1 - 14x + 49​x 2​ ​
​Factors
of 49 Factors
    
of 1 Outer + Inner
___________________________________
​1 and 49 -1 and -1 1(-1) + 49(-1) = -507
7 and 7 -1 and -1 7(-1) + 7(-1) = -143
(7x - 1)(7x - 1)
240 Holt McDougal Algebra 1
83. 1 + 18x + 81​x 2​ ​
​Factors
of 81 Factors
    
of 1 Outer + Inner​
____________________________________
1 and 81 1 and 1 1(1) + 81(1) = 82 7
9 and 9 1 and 1 9(1) + 9(1) = 18 3
(9x + 1)(9x + 1)
3
7. 6​p ​ ​= 2 · 3 · p · p · p
2p = 2 · p
The GCF of 6
​ p 3​ ​and 2p is 2p.
3
8. 12​x ​ ​= 2 · 2 · 3 · x · x · x
18​x 4​ ​= 2 · 3 · 3 · x · x · x · x
The GCF of 12​x 3​ ​and 18​x ​4​is 6​x 3​ ​.
84. 25 + 30x + 9​x ​2​
​Factors
of 9 Factors
    
of 25 Outer + Inner​
____________________________________
1 and 9 1 and 25 1(25) + 9(1) = 34 7
1 and 9 25 and 1 1(1) + 9(25) = 226 7
1 and 9 5 and 5 1(5) + 9(5) = 50 7
3 and 3 1 and 25 3(25) + 3(1) = 78 7
3 and 3 5 and 5 3(5) + 3(5) = 30 3
(3x + 5)(3x + 5)
9. -15 = -1 · 3 · 5
4
20​s ​ ​= 2 · 2 · 5 · s · s · s · s
The GCF of -15 and 20​s 4​ ​is 5.
10. 3a = 3 · a
2
4​b ​ ​= 2 · 2 · b · b
The GCF of 3a and 4​b 2​ ​is 1.
85. 4 + 20x + 25​x ​2​
​Factors
of 25 Factors
    
of 4 Outer + Inner​
____________________________________
1 and 25 1 and 4 1(4) + 25(1) = 29 7
1 and 25 4 and 1 1(1) + 25(4) = 101 7
1 and 25 2 and 2 1(2) + 25(2) = 52 7
5 and 5 1 and 4 5(4) + 5(1) = 25 7
5 and 5 2 and 2 5(2) + 5(2) = 20 3
(5x + 2)(5x + 2)
86. 4 - 12x + 9​x ​2​
​___________________________________
Factors
of 9 Factors
    
of 4 Outer + Inner​
1 and 9 -1 and -4 1(-4) + 9(-1) = -13 7
1 and 9 -4 and -1 1(-1) + 9(-4) = -37 7
1 and 9 -2 and -2 1(-2) + 9(-2) = -20 7
3 and 3 -1 and -4 3(-4) + 3(-1) = -15 7
3 and 3 -2 and -2 3(-2) + 3(-2) = -12 3
(3x - 2)(3x - 2)
87. 3​x 2​ ​+ bx + 2
​___________________________________
Factors
of 3 Factors
    
of 2 Outer + Inner​
1 and 3 1 and 2 1(2) + 3(1) = 5
1 and 3 2 and 1 1(1) + 3(2) = 7
1 and 3 -1 and -2 1(-2) + 3(-1) = -5
1 and 3 -2 and -1 1(-1) + 3(-2) = -7
Possible values of b are -7, -5, 5, and 7.
88. 3​x 2​ ​+ bx - 2
​_
Factors
of 3 Factors
    
of -2 Outer + Inner
___________________________________
​1 and 3 1 and -2 1(-2) + 3(1) = 1
1 and 3 -1 and 2 1(2) + 3(-1) = -1
1 and 3 2 and -1 1(-1) + 3(2) = 5
1 and 3 -2 and 1 1(1) + 3(-2) = -5
Possible values of b are -5, -1, 1, and 5.
89. 5​x ​2​+ bx + 1
​___________________________________
Factors
of 5 Factors
    
of 1 Outer + Inner
​1 and 5 1 and 1 1(1) + 5(1) = 6
1 and 5 -1 and -1 1(-1) + 5(-1) = -6
Possible values of b are -6 and 6.
ready to go on? Section A Quiz
1. 54 = 2 · 3 · 3 · 3 = 2 · 3
​  ​3​
2. 42 = 2 · 3 · 7
2
3. 50 = 2 · 5 · 5 = 2 · 5
​  ​ ​
3
4. 120 = 2 · 2 · 2 · 3 · 5 = ​2 ​ ​· 3 · 5
2
5. 44 = 2 · 2 · 11 = ​2 ​ ​· 11
6. 78 = 2 · 3 · 13
11. The 24 American League games’ balls and 30
National League games’ balls must be divided into
groups of equal size. The number of balls in each
row must be a common factor of 24 and 30.
factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
The GCF of 24 and 30 is 6.
The greatest possible number of balls in each row is
6. Find the number of rows.
24 balls from American League games
​ _______________________________
    
   
 ​= 4 rows
6 balls per row
30 balls from National League games
 ​= 5 rows
    
   
​ ______________________________
6 balls per row
When the greatest possible number of balls is in
each row, there are 9 rows in total.
3
12. 2​d ​ ​= 2 · d · d · d
4d = 2 · 2 · d
The GCF of 2
​ d 3​ ​and 4d is 2d.
3
2​d ​ ​+ 4d
​d ​2​(2d) + 2(2d)
2d(​d 2​ ​+ 2)
2
13.​ m ​ ​= m · m
8​m 5​ ​= 2 · 2 · m · m · m · m · m
The GCF of m
​  2​ ​and 8​m 5​ ​is ​m 2​ ​.
2
5
​m ​ ​- 8​m ​ ​
1(​m ​2​) - 8​m 3​ ​(​m 2​ ​)
​m ​2​(1 - 8​m 3​ ​)
4
14. 12​x ​ ​= 2 · 2 · 3 · x · x · x · x
8​x 3​ ​= 2 · 2 · 2 · x · x · x
4​x 2​ ​= 2 · 2 · x · x
The GCF of 1
​ 2x 4​ ​, 8​x 3​ ​and 4​x ​2​is 4​x 2​ ​.
4
3
12​x ​ ​- 8​x ​ ​- 4​x 2​ ​
4​x ​2​(3​x 2​ ​- 2x - 1)
​_
Factors
of 3 Factors
    
of -1 Outer + Inner
___________________________________
​1 and 3 1 and -1 1(-1) + 3(1) = 2 7
1 and 3 -1 and 1 1(1) + 3(-1) = -2 3
3​x ​2​- 2x - 1 = (x - 1)(3x + 1)
12​x 4​ ​- 8​x 3​ ​- 4​x 2​ ​= 4​x 2​ ​(x - 1)(3x + 1)
2
15. 3​k ​ ​= 3 · k · k
6k = 2 · 3 · k
3 =3
The GCF of 3
​ k 2​ ​, 6k, and 3 is 3.
2
3​k ​ ​+ 6k - 3
​k 2​ ​(3) + 2k(3) - 1(3)
3(​k 2​ ​+ 2k - 1)
241 Holt McDougal Algebra 1
16. sπr = s · π · r
π​r 2​ ​= π · r · r
The GCF of sπr and​πr 2​ ​is πr.
sπr + π​r ​2​
s(πr) + r(πr)
πr(s + r)
27. n(n+ 3) - 4
​n 2​ ​+ 3n - 4
​_________________
Factors
of -4   
Sum​
-1 and 4 3 3
(n - 1)(n + 4)
3
3
2
19. 2​p ​ ​- 6​p ​ ​+ 15 - 5p
(2​p 3​ ​- 6​p 2​ ​) - (5p - 15)
2​p ​2​(p - 3) - 5(p - 3)
(2​p 2​ ​- 5)(p - 3)
3
0
2
18. 3​x ​ ​+ 6​x ​ ​- 4x - 8
(3​x ​3​+ 6​x 2​ ​) - (4x + 8)
3​x ​2​(x + 2) - 4(x + 2)
(3​x 2​ ​- 4)(x + 2)
2
20.​n ​ ​- 6​n ​ ​+ 5n - 30
(​n 3​ ​- 6​n 2​ ​) + (5n - 30)
​n ​2​(n - 6) + 5(n - 6) = (​n 2​ ​+ 5)(n - 6)
2
21.​n ​ ​+ 9n + 20
​ _______________
Factors
of 20   
Sum​
_
1 and 20 21 7
2 and 10 12 7
4 and 5 9 3
(n + 4)(n + 5)
22.​d 2​ ​- 6d - 7
​ ________________
Factors
of -7   
Sum​
_
1 and -7 -6 3
(d + 1)(d - 7)
2
23.​x ​ ​- 6x + 8
​_______________
Factors
of 8   
Sum​
-1 and -8 -9 7
-2 and -4 -6 3
(x - 2)(x - 4)
2
24.​y ​ ​+ 7y - 30
​__________________
Factors
of -30  
Sum​
-1 and 30 29 7
-2 and 15 13 7
-3 and 10 7 3
(y - 3)(y + 10)
​n ​2​+ 3n - 4
n
(n - 1)(n + 4)
2
​0 ​ ​+ 3(0) - 4 = -4
0
(0 + 4)(0 - 1) = -4
2
1
(1 + 4)(1 - 1) = 0
2
2
(2 + 4)(2 - 1) = 6
2
3
(3 + 4)(3 - 1) = 14
2
4
(4 + 4)(4 - 1) = 24
n
17.​w ​3​- 4​w 2​ ​+ w - 4
(​w ​3​- 4​w 2​ ​) + (w - 4)
​w ​2​(w - 4) + 1(w - 4)
(​w 2​ ​+ 1)(w - 4)
1
​1 ​ ​+ 3(1) - 4 = 0
2
​2 ​ ​+ 3(2) - 4 = 6
​3 ​ ​+ 3(3) - 4 = 14
3
​4 ​ ​+ 3(4) - 4 = 24
4
2
28. 2​x ​ ​+ 11x + 5
​___________________________________
Factors
of 2 Factors
    
of 5 Outer + Inner
​1 and 2 1 and 5 1(5) + 2(1) = 7 7
1 and 2 5 and 1 1(1) + 2(5) = 11 3
(x + 5)(2x + 1)
29. 3​n 2​ ​+ 16n + 21
​____________________________________
Factors
of 3 Factors
    
of 21 Outer + Inner​
1 and 3 1 and 21 1(21) + 3(1) = 24 7
1 and 3 21 and 1 1(1) + 3(21) = 64 7
1 and 3 3 and 7 1(7) + 3(3) = 16 3
(n + 3)(3n + 7)
30. 5​y 2​ ​- 7y - 6
​_
Factors
of 5 Factors
    
of -6 Outer + Inner
___________________________________
​1 and 5 1 and -6 1(-6) + 5(1) = -1 7
1 and 5 -1 and 6 1(6) + 5(-1) = 1 7
1 and 5 2 and -3 1(-3) + 5(2) = 7 7
1 and 5 -2 and 3 1(3) + 5(-2) = -7 3
(y - 2)(5y + 3)
31. 4​g ​2​- 10g + 6
​___________________________________
Factors
of 4 Factors
    
of 6 Outer + Inner
​1 and 4 -1 and -6 1(-6) + 4(-1) = -10 3
(g - 1)(4g - 6)
2(g - 1)(2g - 3)
32. 6​p 2​ ​- 18p - 24
​____________________________________
Factors
of 6 Factors
    
of -24 Outer + Inner​
1 and 6 1 and -24 1(-24) + 6(1) = -183
(p + 1)(6p - 24)
6(p + 1)(p - 4)
25.​k ​2​- 6k + 5
​_______________
Factors
of 5   
Sum
​-1 and -5 -6 3
(k - 1)(k - 5)
2
26.​c ​ ​- 10c + 24
​________________
Factors
of 24   
Sum
​-1 and -24 -25 7
-2 and -12 -14 7
-3 and -8 -11 7
-4 and -6
-10 3
(c - 4)(c - 6)
242 Holt McDougal Algebra 1
33. 12​d 2​ ​+ 7d - 12
​Factors
of 12     
Factors of -12 Outer + Inner​
_____________________________________
7
1 and 12 1 and -12 1(-12) + 12(1) = 0
7
1 and 12 -1 and 12 1(12) + 12(-1) = 0
1 and 12 2 and -6 1(-6) + 12(2) = 18 7
1 and 12 -2 and 6 1(6) + 12(-2) = -18 7
2 and 6 1 and -12 2(-12) + 6(1) = -18 7
2 and 6 -1 and 12 2(12) + 6(-1) = 18 7
7
2 and 6 2 and -6 2(-6) + 6(2) = 0
7
2 and 6 -2 and 6 2(6) + 6(-2) = 0
2 and 6 3 and -4 2(-4) + 6(3) = 10 7
2 and 6 -3 and 4 2(4) + 6(-3) = -10 7
3 and 4 1 and -12 3(-12) + 4(1) = -32 7
3 and 4 -1 and 12 3(12) + 4(-1) = 32 7
3 and 4 2 and -6 3(-6) + 4(2) = -10 7
3 and 4
-2 and 6 3(6) + 4(-2) = 10 7
7
3 and 4 3 and -4 3(-4) + 4(3) = 0
7
3 and 4 -3 and 4 3(4) + 4(-3) = 0
3 and 4 4 and -3 3(-3) + 4(4) = 7 3
(3d + 4)(4d - 3)
think and discuss
1. 1 - x​  4​ ​
2
​1 ​2​- ​(​x 2​ ​) ​
​(1 + x​  2​ ​)(1 - x​  2​ ​)
a = 1, b = x​  ​2​
3.
2
2.​x ​ ​+ 8x + 16
​x 2​ ​+ 2(x)(4) + ​4 ​2​
​(x + 4) ​2​
a = x, b = 4
Special Product
Factored Form
Perfect-square trinomial with positive
coefficient of middle term:
x 2 + 2x + 1
(x + 1) 2
Perfect-square trinomial with negative
coefficient of middle term:
x 2 - 2x + 1
(x - 1) 2
Difference of two squares:
x2 - 1
(x - 1)(x + 1)
Exercises
Guided Practice
7-5 factoring SPECIAL PRODUCTS
1. Yes
​x 2​ ​- 4x + 4
​x 2​ ​- 2(x)(2) + 2
​  ​2​
​(x - 2) ​2​
check it out!
2. No, the trinomial is not a perfect square because the
last term of the trinomial is not positive.
1a. Yes, the trinomial is a perfect square.​
x 2​ ​+ 4x + 4
​x ​2​+ 2(x)(2) + 2
​  ​2​
​(x + 2) ​2​
3. Yes
9​x ​2​- 12x + 4
​(3x) ​2​- 2(3x)(2) + 2
​  ​2​
2
​(3x - 2) ​ ​
b. Yes, the trinomial is a perfect square.
2
​x ​ ​- 14x + 49
​x ​2​- 2(x)(7) + 7
​  ​2​
2
​(x - 7) ​ ​
5. Yes
​x ​2​- 6x + 9
​x ​2​- 2(x)(3) + 3
​  ​2​
​(x - 3) ​2​
c. No, the trinomial is not a perfect square.
​9x ​2​= ​(3x) ​2​, 4 = 2
​  ​2​, but -6x ≠ 2(3x)(2)
6. No, the trinomial is not a perfect square because the
last term of the trinomial is not positive.
34. (4x + 2) cm
2
2. 9​x ​ ​+ 6x + 1
​(3x) ​2​+ 2(3x)(1) + 1
​  ​2​
2
​(3x + 1) ​ ​
The perimeter of each sheet is 4(3x + 1) m.
x = 3, 4(3x + 1) = 4(3(3) + 1) = 40
The perimeter is 40 m when x = 3 m.
3a. Yes, the binomial is a difference of two squares.
1 - 4​x ​2​
​1 ​2​- ​(2x) ​2​
(1 + 2x)(1 - 2x)
1 - 4​x ​2​= (1 + 2x)(1 - 2x)
b.​Yes, the binomial is a difference of two squares.
8
6
p ​ ​- 49​q ​ ​
2
4 2
​(p
​  ​ ​) ​ ​- ​(7​q 3​ ​) ​ ​
4
3
4
(​p ​ ​+ 7​q ​ ​)(​p ​ ​- 7​q 3​ ​)
​p ​8​- 49​q 6​ ​= (​p 4​ ​+ 7​q 3​ ​)(​p 4​ ​- 7​q 3​ ​)
c. No, the binomial is not a difference of two squares
5
because 4​y ​ ​is not a perfect square.
4. Yes
​x ​2​+ 2x + 1
​x 2​ ​+ 2(x)(1) + ​1 ​2​
2
(x + 1​) ​ ​
2
7.​x ​ ​+ 24x + 144
​x ​2​+ 2(x)(12) + 1
​ 2 ​2​
2
​(x + 12) ​ ​
The length and width are both (x + 12) yd.
The perimeter of the park is 4(x + 12) yd.
x = 10; 4(x + 12) = 4(10 + 12) = 88
The perimeter of the park is 88 yd when x = 10 yd.
8. Yes
1 - 4​x ​2​
​1 ​2​- ​(2x) ​2​
(1 + 2x)(1 - 2x)
9. Yes
​s ​2​- ​4 ​2​
(s + 4)(s - 4)
10. Yes
81​x ​2​- 1
​(9x) ​2​- ​1 ​2​
(9x + 1)(9x - 1)
11. Yes
4​x ​4​- 9​y 2​ ​
2 2
2
(​ 2​x ​ ​) ​ ​- ​(3y) ​ ​
(2​x ​2​+ 3y)(2​x 2​ ​- 3y)
12. No, the binomial is not a difference of two squares
because 50 is not a perfect square.
13. Yes
​x ​6​- 9
3 2
2
(​ x​  ​ ​) ​ ​- ​3 ​ ​
(​x ​3​+ 3)(​x 3​ ​- 3)
243 Holt McDougal Algebra 1
practice and problem solving
34. Perfect-square trinomial 35. Difference of 2 squares
2
2
​(x​  7​ ​) ​ ​- ​(12) ​2​
​49x ​ ​- 70x + 25
2
2
​(7x) ​ ​- 2(7x)(5) + 5
​  ​ ​
(​x ​7​+ 12)(​x 7​ ​- 12)
​(7x - 5) ​2​
14. Yes
4​x 2​ ​- 4x + 1
​(2x) ​2​- 2(2x)(1) + 1
​  ​2​
2
​(2x - 1) ​ ​
15. No, the trinomial is not a perfect square because the
last term of the trinomial is not positive.
16. Yes
36​x ​2​- 12x + 1
​(6x) ​2​- 2(6x)(1) + 1
​  ​2​
2
​(6x - 1) ​ ​
37. Possible answer: multiply a binomial by itself.
Choose 2 perfect squares, find 2 times the product
of their square roots, and then write these
3 expressions as a sum.
17. No, the trinomial is not a perfect square.
2
2
​  ​2​, but 10x ≠ 2(5x)(2)
25​x ​ ​= ​(5x) ​ ​, 4 = 2
38.​x ​2​- 22x + 121
​x 2​ ​- 2(x)(11) + 1
​ 1 ​2​
2
​(x - 11) ​ ​
b = -11
19. Yes
​16x ​2​- 40x + 25
(4x​) ​2​- 2(4x)(5) + 5
​  ​2​
2
​(4x - 5) ​ ​
18. Yes
​9x ​2​+ 18x + 9
​(3x) ​2​+ 2(3x) + ​3 ​2​
​(3x + 3) ​2​
9(x + 1​)2​ ​
2
2
b. ℓ =
(x + 5) ft
w = (x - 5) ft
5 feet were added to the length and subtracted
from the width.
c. ℓ = (x + 5) = (8 + 5) = 13 ft
w = (x - 5) = (8 - 5) = 3 ft
1a. 25​z ​2​- 40z + 16
4
​(5z) ​2​- 2(5z)(4) + 4
​  ​2​
2
​(5z - 4) ​ ​
The length of a side of the square is 5z - 4.
22. Yes
25​m ​2​- 16​n 2​ ​
​(5m) ​2​- ​(4n) ​2​
(5m + 4n)(5m - 4n)
b. The perimeter of the square is 4(5z - 4) = 20z - 16.
c. z = 3
5z - 4 = 5(3) - 4 = 11
4(5z - 4) = 4(11) = 44
​(5z - 4) ​2​= ​(11) ​2 ​= 121
When z = 3, the length of a side is 11, the
perimeter is 44, and the area is 121.
23. No, the binomial is not a difference of two squares
because 4x and 9y are not perfect squares.
24. Yes
25.​Yes
49​p ​12​- 9​q 6​ ​
9 ​2​- 100​x 4​ ​
2
6 2
3 2
​(7​p ​ ​) ​ ​- ​(3​q ​ ​) ​ ​
​9 ​2​-​ (10​x 2​ ​) ​ ​
6
3
6
3
2
(7​p ​ ​+ 3​q ​ ​)(7​p ​ ​- 3​q ​ ​)
(9 + 10​x ​ ​)(9 - 10​x 2​ ​)
2a. The area of the larger rectangle is 3x(x) = 3​x ​2​.
4
The area of the smaller rectangle is 3y(y) = 3​y 2​ ​.
26. No, the binomial is not a difference of two squares
because x​  ​3​and ​y 3​ ​are not perfect squares.
2
2
2
2
c. 3​x ​ ​- 3​y ​ ​
3(​x 2​ ​- ​y 2​ ​)
3(x + y)(x - y)
2
28. 9​x ​ ​= ​(3x) ​ ​, 25 = ​5 ​ ​
a = 3x, b = 5, and 2ab = 2(3x)(5) = 30x
9​x ​2​+ 30x + 25
3a. x = -5
4
​x ​2​+ 10x + 25 = (​ -5) ​2​+ 10(-5) + 25 = 0
​(x + 5) ​2​= ​(-5 + 5) ​2​= 0
​(x - 5) ​2​= ​(-5 - 5) ​2​= 100
​x ​2​- 10x + 25 = (​ -5) ​2​- 10(-5) + 25 = 100
​x ​2​- 25 = ​(-5) ​2​- 25 = 0
29. 36y = 2(2y)(9)
a = 2y and ​a 2​ ​= ​(2y) ​2​= 4​y 2​ ​
4​y ​2​- 36y + 81
30. Perfect-square trinomial 31. Difference of 2 squares
100​x 2​ ​- 81​y 2​ ​
​x 2​ ​- 8x + 16
2
2
​x ​ ​- 2(x)(4) + 4
​(10x) ​2​- ​(9y) ​2​
​  ​ ​
2
(10x + 9y)(10x - 9y)
​(x - 4) ​ ​
32. Perfect-square trinomial 33. Difference of 2 squares
4​r 6​ ​- 25​s 6​ ​
36​x ​2​+ 24x + 4
2
2
2
2
​(6x) ​ ​+ 2(6x)(2) + 2
​(2​r ​3​) ​ ​- ​(5​s 3​ ​) ​ ​
​  ​ ​
2
​(6x + 2) ​ ​
(2​r ​3​+ 5​s 3​ ​)(2​r ​3​- 5​s 3​ ​)
2
b. The area of the green region is 3​x ​ ​- 3​y ​ ​.
27. 14x = 2(x)(7)
b = 7 and ​b 2​ ​= ​7 ​2​= 49
​x ​2​+ 14x + 49
2
39. 256 = 1
​ 6 ​2​
a = x, b = 16
2ab = 2(x)(16) = 32x
c = 32
0a.​x ​ ​- 25
4
​x 2​ ​- ​5 ​2​
(x + 5)(x - 5)
20. 4​x ​ ​- 44x + 121
​(2x) ​2​- 2(2x)(11) + ​11 ​2​
​(2x - 11) ​2
​The length and width are both (2x - 11) mm.
The perimeter of the rectangle is 4(2x - 11) mm.
x = 41, 4(2x - 11) = 4(2(41) - 11) = 4(71) = 284
The perimeter of the rectangle is 284 mm when
x = 41mm.
21. Yes
​1 ​2​- 4​x 2​ ​
​1 ​2​- ​(2x) ​2​
(1 + 2x)(1 - 2x)
36. Possible answer: they are similar in that the first
and last terms of each are perfect squares. They
are different in that a perfect-square trinomial has
3 terms and a difference of 2 squares has 2 terms.
b. x = -1
2
2
​x ​ ​+ 10x + 25 = (​ -1) ​ ​+ 10(-1) + 25 = 16
2
2
​(x + 5) ​ ​=​ (-1 + 5) ​ ​= 16
​(x - 5) ​2​= ​(-1 - 5) ​2​= 36
​x ​2​- 10x + 25 = (​ -1) ​2​- 10(-1) + 25 = 36
​x ​2​- 25 = ​(-1) ​2​- 25 = -24
244 Holt McDougal Algebra 1
c. x = 0
​x ​2​+ 10x + 25 = ​(0) ​2​+ 10(0) + 25 = 25
​(x + 5) ​2​= ​(0 + 5) ​2​= 25
​(x - 5) ​2​= ​(0 - 5) ​2​= 25
​x ​2​- 10x + 25 = ​(0) ​2​- 10(0) + 25 = 25
​x ​2​- 25 = ​(0) ​2​- 25 = -25
51a. a = 2, b = (v + 2)
2
b. 4 - (​ v + 2) ​ ​
(2 + (v + 2))(2 - (v - 2))
(v + 4)(-v)
-​v ​2​- 4v
52.​x 3​ ​- 1
​x 3​ ​- ​1 ​3​
a = x, b = 1
(x - 1)(​x 2​ ​+ (x)(1) + 1
​  ​2​)
2
(x - 1)(​x ​ ​+ x + 1)
d. x = 1
2
2
​x ​ ​+ 10x + 25 = ​(1) ​ ​+ 10(1) + 25 = 36
2
2
​(x + 5) ​ ​= ​(1 + 5) ​ ​= 36
​(x - 5) ​2​= ​(1 - 5) ​2​= 16
​x ​2​- 10x + 25 = ​(1) ​2​- 10(1) + 25 = 16
​x ​2​- 25 = ​(1) ​2​- 25 = -24
3
53. 27​y ​ ​- 64
​(3y) ​3​- ​4 ​3​
a = 3y, b = 4
(3y - 4)(​(3y) ​2​+ (3y)(4) + 4
​  ​2​)
(3y - 4)(9​y ​2​+ 12y + 16)
e. x = 5
2
2
​x ​ ​+ 10x + 25 = ​(5) ​ ​+ 10(5) + 25 = 100
2
2
​(x + 5) ​ ​= ​(5 + 5) ​ ​= 100
​(x - 5) ​2​= ​(5 - 5) ​2​= 0
​x ​2​- 10x + 25 = ​(5) ​2​- 10(5) + 25 = 0
​x ​2​- 25 = ​(5) ​2​- 25 = 0
6
44. Columns 1 and 2 have equivalent values because
2
2
​x ​ ​+ 10x + 25 =​ (x + 5) ​ ​.
Columns 3 and 4 have equivalent values because​
(x - 5) ​2​= ​x 2​ ​- 10x + 25.
45. The missing labels are (a + b) and (a - b).
4
54.​n ​ ​- 8
3
​(​n 2​ ​) ​ ​- ​2 ​3​
a = ​n 2​ ​, b = 2
2
2 2
2
2
​  ​ ​)
(​n ​ ​- 2)(​(​n ​ ​) ​ ​+ (​n ​ ​)(2) + 2
(​n ​2​- 2)(​n 4​ ​+ 2​n 2​ ​+ 4)
7-6 choosing a factoring method
46. Student A is incorrect because (5x)(5x) ≠ 25​x ​ ​, and
(-3)(3) ≠ 9​y 2​ ​.
check it out!
test prep
1a. Yes, 5​x 2​ ​(x - 1) is completely factored.
47. C;
​x 2​ ​- 2xy + ​y 2​ ​
​x ​2​- 2(x)(y) + y​  2​ ​
​(x - y) ​2​
x = 0, y = 0; (​ x - y)​2​= ​(0 - 0) ​2​= 0
x = -1, y = -1; ​(x - y) ​2​= (​ -1 + 1) ​2​= 0
x = 1, y = 1; (​ x - y) ​2​= ​(1 - 1) ​2​= 0
x = 1, y = -1; (​ x - y) ​2​= ​(1 + 1) ​2​= 4
48. J;
2
4​x ​ ​+ 20x + 25​
(2x) ​2 ​+ 2(2x)(5) + 5
​  ​2​
2
​(2x + 5) ​ ​
2
challenge and extend
b. 9​x ​2​- 4
​(3x) ​2​- ​2 ​2​
(3x + 2)(3x - 2)
4
c. Possible answer: ​x ​ ​- 1
2
​(x​  ​2​) ​ ​- ​1 ​2​
(​x ​2​+ 1)(​x 2​ ​- 1)
(​x ​2​+ 1)(​x 2​ ​- ​1 ​2​)
(​x ​2​+ 1)(x + 1)(x - 1)
3
2
2
3
2a. 4​x ​ ​+ 16​x ​ ​+ 16x
4x(​x 2​ ​+ 4x + 4)
4x​(x + 2) ​2​
b. 2​x ​ ​y - 2​y ​ ​
2y(​x 2​ ​- ​y 2​ ​)
2y(x + y)(x - y)
2
49.​x ​ ​- 18x + 81
​x ​2​- 2(x)(9) + 9
​  ​2​
2
​(x - 9) ​ ​
x = 10, ​x ​2​- 18x + 81 = ​(x - 9) ​2​= ​(10 - 9) ​2​= 1
0a. 81​x ​4​- 16
5
2
​(9​x ​2​) ​ ​- ​4 ​2​
2
(9​x ​ ​+ 4)(9​x 2​ ​- 4)
b. No, (4x + 4)(x + 1) is not completely factored.
(4x + 4)(x + 1)
4(x + 1)(x + 1)
​4(x + 1) ​2​
3a. 3​x ​ ​+ 7x + 4
​_
Factors
of 3 Factors
    
of 4 Outer + Inner​
_________________________________
1 and 3 1 and 4 1(4) + 3(1) = 7 3
(x + 1)(3x + 4)
5
4
3
6
5
4
b. 2​p ​ ​+ 10​p ​ ​- 12​p ​ ​
2​p ​3​(​p 2​ ​+ 5p - 6)
​Factors
of -6   
Sum​
_________________
-1 and 6 5 3
2​p ​3​(p - 1)(p + 6)
c. 9​q ​ ​+ 30​q ​ ​+ 24​q ​ ​
3​q ​4​(3​q 2​ ​+ 10q + 8)
​___________________________________
Factors
of 3 Factors
    
of 8 Outer + Inner
​1 and 3 1 and 8 1(8) + 3(1) = 11 7
1 and 3 8 and 1 1(1) + 3(8) = 25 7
1 and 3 2 and 4 1(4) + 3(2) = 10 3
3​q 4​ ​(q + 2)(3q + 4)
4
d. 2​x ​ ​+ 18
2(​x 4​ ​) + 2(9)
2(​x 4​ ​+ 9)
245 Holt McDougal Algebra 1
2
think and discuss
1. Possible answer: (​x 2​ ​+ 1)(​x 2​ ​- 1)
2
2
2. Possible answer:​x ​ ​+ 1; x​  ​ ​+ x + 1
3.
Factoring Methods
Polynomial
Method
1. 16x 4 - 25y 8
A. Factoring out the GCF
2. x 2 + 1 0 x + 25
B. Factoring by grouping
3. 9t 2 + 27t + 18t 4
C. Unfactorable
4. a 2 + 3a - 7a - 21
D. Difference of two squares
5. 100b 2 + 81
E. Perfect-Square trinomial
exercises
1. Yes, 3x(9​x 2​ ​+ 1) is completely factored.
2. No
3
2
2(4​x ​ ​- 3​x ​ ​- 8x)
2
2x(4​x ​ ​- 3x - 8)
3
3. Yes, 2​k ​ ​(4 - ​k ​ ​) is completely factored.
4. Yes, (2x + 3)(3x - 5) is completely factored.
5. No
2
4(4​p ​ ​- 1)
4[​(2p) ​2​- ​1 ​2​]
4(2​p ​ 2​+ 1)(2​p ​ 2​- 1)
3
3
7. 3​x ​ ​- 12​x ​ ​
3​x ​3​(​x 2​ ​- 4)
3​x ​3​(x + 2)(x - 2)
3
2
8. 4​x ​ ​+ 8​x ​ ​+ 4x
4x(​x ​2​+ 2x + 1)
4x​(x + 1) ​2​
2
9. 8p​q ​ ​+ 8pq + 2p
2p(4​q 2​ ​+ 4q + 1)
2p​(2q + 1) ​2​
2
10. 18r​s ​ ​- 2r
2r(9​s 2​ ​- 1)
2r(3s + 1)(3s - 1)
5
3
11. m​n ​ ​- ​m ​ ​n
mn(​n ​4​- ​m 2​ ​)
mn(​n ​2​+ m)(​n 2​ ​- m)
2
12. 2​x ​ ​y - 20xy + 50y
2y(​x 2​ ​- 10x + 25)
2y​(x - 5) ​2​
4
3
2
13. 6​x ​ ​- 3​x ​ ​- 9​x ​ ​
3​x ​2​(2​x 2​ ​- x - 3)
​_
Factors
of 2 Factors
    
of -3 Outer + Inner​
___________________________________
1 and 2 1 and -3 1(-3) + 2(1) = -1 3
3​x ​2​(x + 1)(2x - 3)
3
2
5
4
3
16. 7​x ​ ​+ 21​x ​ ​- 28​x ​ ​
7​x ​3​(​x 2​ ​+ 3x - 4)
​Factors
of -4   
Sum​
_________________
-1 and 4 3 3
7​x ​3​(x - 1)(x + 4)
17. 2​z ​ ​+ 11z + 6
​___________________________________
Factors
of 2 Factors
    
of 6 Outer + Inner​
1 and 2 1 and 6 1(6) + 2(1) = 8 7
1 and 2 6 and 1 1(1) + 2(6) = 13 7
1 and 2 2 and 3 1(3) + 2(2) = 7 7
1 and 2 3 and 2 1(2) + 2(3) = 8 7
2​z 2​ ​+ 11z + 6 cannot be factored.
2
2
18. 9​p ​ ​- ​q ​ ​+ 3p + q
(9​p ​2​- ​q 2​ ​) + (3p + q)
(3p - q)(3p + q) + 1(3p + q)
(3p - q + 1)(3p + q)
practice and problem solving
2
6. Yes, a(​a ​ ​+ 2ab + ​b ​ ​) is completely factored.
5
5
15.​p ​ ​+ 3​p ​ ​+ ​p ​ ​+ 3
(​p ​5​+ 3​p 3​ ​) + (​p 2​ ​+ 3)
​p ​3​(​p 2​ ​+ 3) + 1(​p 2​ ​+ 3)
(​p ​3​+ 1)(​p 2​ ​+ 3)
2
guided practice
2
14. 3​y ​ ​+ 14y + 4
​___________________________________
Factors
of 3 Factors
    
of 4 Outer + Inner
​1 and 3 1 and 4 1(4) + 3(1) = 7 7
1 and 3 4 and 1 1(1) + 3(4) = 13 7
1 and 3 2 and 2 1(2) + 3(2) = 8 7
3​y 2​ ​+ 14y + 4 cannot be factored.
19. No
2x(​y 3​ ​- 4​y 2​ ​+ 5y)
2xy(​y ​2​- 4y + 5)
20. No
6
2r(25​r ​ ​- 36)
3 2
2
2r[​(5​r ​ ​) ​ ​-​ 6 ​ ​]
2r(5​r ​3​+ 6)(5​r 3​ ​- 6)
21. No
3​n ​2​(​n 2​ ​- 25)
3​n ​2​(​n 2​ ​- ​5 ​2​)
3​n ​2​(n + 5)(n - 5)
22. Yes, 2m(m + 1)(m + 4) is completely factored.
2
2
23. Yes, 2​y ​ ​(4​x ​ ​+ 9) is completely factored.
2
24. Yes, 4(7g + 9​h ​ ​) is completely factored.
3
2
25. -4​x ​ ​+ 24​x ​ ​- 36x
-4x(​x ​2​- 6x + 9)
-4x​(x - 3) ​2​
2
4
26. 24​r ​ ​- 6​r ​ ​
6​r ​2​(4 - r​  2​ ​)
6​r ​2​(2 + r)(2 - r)
2
27. 5​d ​ ​- 60d + 135
5(​d 2​ ​- 12d + 27)
​__________________
Factors
of 27   
   Sum​
-1 and -27 -28 7
-3 and -9 -12 3
5(d - 3)(d - 9)
28. 4​y 8​ ​+ 36​y 7​ ​+ 81​y 6​ ​
​y ​6​(4​y 2​ ​+ 36y + 81)
​y ​6​​(2y + 9) ​2​
246 Holt McDougal Algebra 1
29. 98​x 3​ ​- 50x​y 2​ ​
2x(49​x ​2​- 25​y 2​ ​)
2x(7x + 5y)(7x - 5y)
3
40. Let a be the number of apples on the tree.
3​a 2​ ​- 22a + 35
​___________________________________
Factors
of 3 Factors
    
of 35 Outer + Inner​
1 and 3 -1 and -35 1(-35) + 3(-1) = -38 7
1 and 3 -35 and -1 1(-1) + 3(-35) = -1067
1 and 3 -5 and -7 1(-7) + 3(-5) = -22 3
(a - 5)(3a - 7)
2
30. 4​x ​ ​y - 4​x ​ ​y - 8xy
4xy(​x ​2​- x - 2)
​Factors
of -2     Sum​
  
___________________
1 and -2 -1 3
4xy(x + 1)(x - 2)
2
31. 5​x ​ ​- 10x + 14
​___________________________________
Factors
of 5 Factors
    
of 14 Outer + Inner
​1 and 5 -1 and -14 1(-14) + 5(-1) = -197
1 and 5 -14 and -1 1(-1) + 5(-14) = -717
1 and 5 -2 and -7 1(-7) + 5(-2) = -177
1 and 5 -7 and -2 1(-2) + 5(-7) = -377
5​x 2​ ​- 10x + 14 cannot be factored.
2
​ 36​y 2​ ​cannot be factored.
32. 121​x ​ +
4
33.​p ​ ​- 16
(​p 2​ ​+ 4)(​p 2​ ​- 4)
(​p ​2​+ 4)(p + 2)(p - 2)
6
5
4
34. 4​m ​ ​- 30​m ​ ​+ 36​m ​ ​
2​m ​4​(2​m 2​ ​- 15m + 18)
​___________________________________
Factors
of 2 Factors
    
of 18 Outer + Inner​
1 and 2 -1 and -181(-18) + 2(-1) = -207
1 and 2 -18 and -1 1(-1) + 2(-18) = -377
1 and 2 -2 and -9 1(-9) + 2(-2) = -137
1 and 2 -9 and -2 1(-2) + 2(-9) = -207
1 and 2 -3 and -6 1(-6) + 2(-3) = -127
1 and 2 -6 and -3 1(-3) + 2(-6) = -153
2​m 4​ ​(m - 6)(2m - 3)
3
2
35. 2​k ​ ​+ 3​k ​ ​+ 6k + 9
(2​k ​3​+ 3​k 2​ ​) + (6k + 9)
​k ​2​(2k + 3) + 3(2k + 3)
(​k 2​ ​+ 3)(2k + 3)
(​k 2​ ​+ 3)(2k + 3)
4
36. a​b ​ ​- 16a
a(​b ​4​- 16)
a(​b 2​ ​+ 4)(​b 2​ ​- 4)
a(​b ​2​+ 4)(b + 2)(b - 2)
37. Let x be Ella’s age.
2
​x ​ ​+ 12x + 36
​x ​2​+ 2(x)(6) + 6
​  ​2​
2
​(x + 6) ​ ​
38. Let d be the distance from point A to point B.
2
​d ​ ​- 81
​d ​2​- ​9 ​2​
(d + 9)(d - 9)
39. Let s be the number of seconds Bob can hold.
​s ​2​- 16s + 28
​________________
Factors
of 28   
Sum
​-1 and -28 -29 7
-2 and -14 -16 3
(s - 2)(s - 14)
41. Let b be Beth’s score.
​b ​2​- 49
​b 2​ ​- ​7 ​2​
(b + 7)(b - 7)
2
42. -5​t ​ ​+ 30t + 1
-1(5​t ​2​- 30t - 1)
​_
Factors
of 5 Factors
    
of -1 Outer + Inner
___________________________________
​1 and 5 1 and -1 1(-1) + 5(1) = 4 7
1 and 5 -1 and 1 1(1) + 5(-1) = -4 7
-5​t 2​ ​+ 30t + 1 is fully factored.
43. The next step is to check for a pattern, such as
a perfect-square trinomial, or a difference of
2 squares.
2
44. 12​(x + 1) ​ ​+ 60(x + 1) + 75
3[​4(x + 1) ​2​+ 20(x + 1) + 25]
3[(​2(x + 1)) ​2​+ 2(2(x + 1))(5) + 5
​  ​2​]
2
3​[2(x + 1) + 5] ​ ​
3​(2x + 7) ​2
​45.​(2x + 3) ​2​- ​(x - 4) ​2
​[(2x + 3) + (x - 4)][(2x + 3) - (x - 4)]
(3x - 1)(x + 7)
2
46. 45x​(x - 2) ​ ​+ 60x(x - 2) + 20x
5x[9​(x - 2) ​2​+ 12(x - 2) + 4]
5x​[3(x - 2)) ​2​+ 2(3(x - 2))(2) + 2
​  ​2​]
2
5x​[3(x - 2) + 2] ​ ​
5x​(3x - 4) ​2​
2
2
47.​(3x - 5) ​ ​- ​(y + 2) ​ ​
[(3x - 5) + (y + 2)][(3x - 5) - (y + 2)]
(3x + y - 3)(3x - y - 7)
8a.​x ​2​+ 2x - 15
4
​Factors
of -15  
Sum​
__________________
-1 and 15
14 7
-3 and 5
2 3
(x - 3)(x + 5)
b.
x+5
x-3
c. x = 7
ℓ = x + 5 = 7 + 5 = 12 ft
w = x - 3 = 7 - 3 = 4 ft
49. Method 1: 4​x 2​ ​- 100 = 4(​x 2​ ​- 25)
= 4(x + 5)(x - 5)
Method 2: 4​x ​2​- 100 = (2x + 10)(2x - 10)
= 2(x + 5) · 2(x - 5)
= 4(x + 5)(x - 5)
247 Holt McDougal Algebra 1
50. 2​x 2​ ​+ 5xy + 3​y 2​ ​
​___________________________________
Factors
of 2 Factors
    
of 3 Outer + Inner
​1 and 2 1 and 3 1(3) + 2(1) = 5 3
(x + y)(2x + 3y)
x = -10.1, y = 10.05
2​x 2​ ​+ 5xy + 3​y 2​ ​= (x + y)(2x + 3y)
= approx. 0
2
61.​h 2​ ​+ ​h 8​ ​+ ​h 6​ ​+ ​h 4​ ​
​h ​2​(1) + h
​  2​ ​(​h 6​ ​) + ​h 2​ ​(​h 4​ ​) + ​h 2​ ​(​h 2​ ​)
2 6
​h ​ ​(​h ​ ​+ ​h 4​ ​+ ​h 2​ ​+ 1)
​h ​2​[(​h 6​ ​+ ​h 4​ ​) + (​h 2​ ​+ 1)]
​h ​2​[​h 4​ ​(​h 2​ ​+ 1) + (​h 2​ ​+ 1)]
​h ​2​(​h 4​ ​+ 1)(​h 2​ ​+ 1)
n+2
​+ ​x n​ + 1​+ ​x n​ ​
62.​ x ​
n 2
​x ​ ​(​x ​ ​) + ​x n​ ​(x) + x​  n​ ​(1)
​x ​n​(​x 2​ ​+ x + 1)
2
51. Possible answer: (​ 2x - 1) ​ ​≠ 4​x ​ ​- 4x - 1
6
53.​(a + b) ​8​
52.​(a + b) ​ ​
7
54.​(a + b) ​ ​
test prep
55. C;
6​x 2​ ​+ 7x - 10
​Factors
of 6      
Factors of -10 Outer + Inner
_____________________________________
​1 and 6 1 and -10 1(-10) + 6(1) = -47
1 and 6 -1 and 10 1(10) + 6(-1) = 4 7
1 and 6 2 and -5 1(-5) + 6(2) = 7 3
(x + 2)(6x - 5)
56. H;
16​x 12
​ ​- 256
16(​x 12
​ ​- 16)
16(​x 6​ ​+ 4)(​x 6​ ​- 4)
16(​x ​6​+ 4)(​x 3​ ​+ 2)(​x 3​ ​- 2)
2
2
9a. 72π​p ​ ​+ 48π​p ​ ​+ 8π
5
8p(9π​p ​2​+ 6πp + π)
8p[π(9​p 2​ ​+ 6p + 1)]
8p[π​(3p + 1) ​2​]
b. The radius of the cylinder is (3p + 1) cm.
c. 3p + 1 = 4
3p = 3
p=1
h = 8p = 8(1) = 8 cm
V = 8p[π​(3p + 1) ​2​] = 8[π(​4) ​2​] = 128π ​cm ​3​
7
3
5
4
60.​g ​ ​+ ​g ​ ​+ ​g ​ ​+ ​g ​ ​
​g ​3​(​g 4​ ​) + ​g 3​ ​(1) + g
​  3​ ​(​g 2​ ​) + g
​  3​ ​(g)
3 4
2
​g ​ ​(​g ​ ​+ ​g ​ ​+ g + 1)
2. Yes
4​x ​2​- 20x + 25
(​2x) ​2​- 2(2x)(5) + ​5 ​2​
(​2x - 5) ​2​
3. No, (​x ​2​+ 3x + 9) is not a perfect square because
3x ≠ 2(x)(3).
4. No, - 4x ≠ 2(x)(2)
b. The polynomial could be factored by finding factors
of 8 and factors of 18 that would result in 24 as
the sum of the outer and inner products. Then one
binomial would need to be factored again.
3
b. V = w(w + 5)(w + 9)
= (​w ​2​+ 5w)(w + 9)
= ​w ​3​+ 9​w 2​ ​+ 5​w 2​ ​+ 45w
= ​w ​3​+ 14​w 2​ ​+ 45w
1. Yes
​x 2​ ​+ 8x + 16
​x 2​ ​+ 2(x)(4) + 4
​  ​2​
2
(x +​ 4) ​ ​
8a. 8​x ​ ​+ 24​x ​ ​+ 18x
5
2x(4​x ​2​+ 12x + 9)
2x[​(2x) ​2​+ 2(2x)(3) + 3
​  ​2​]
2
2x​(2x + 3) ​ ​
First factor out the GCF of 8​x 3​ ​, 24​x 2​ ​and 18x,
which is 2x; then use the pattern for a perfectsquare trinomial.
challenge and extend
4a. h = w + 5
6
ℓ = w + 9
ready to go on? Section B Quiz
57. C
3
n + 5
​+ ​x n + 4
​
​+ ​x n + 3
​
​
63.​x ​
​x ​n + 3​(​x 2​ ​) + ​x n + 3
​
​(x) + x​  n + 3
​
​(1)
​x ​n + 3​(​x 2​ ​+ x + 1)
5. Yes
2
9​x ​ ​- 12x + 4
​(3x) ​2​- 2(3x)(2) + 2
​  ​2​
2
​(3x - 2) ​ ​
2
6. No, (​x ​ ​- 12x - 36) is not a perfect square because
the last term is not positive.
2
7.​x ​ ​+ 20x + 100
​x 2​ ​+ 2(x)(10) + 1​0 ​2​
​(x + 10) ​2​
ℓ = w = (x + 10) ft
The perimeter of a window is 4(x + 10) ft.
x = 4, 4(x + 10) = 4(4 + 10) = 56
The perimeter of a window is 56 ft when x = 4 ft.
8. Yes
​x ​2​- 121
​x 2​ ​- 1​1 ​2​
(x + 11)(x - 11)
9. No, (4​t 2​ ​- 20) is not a
difference of 2 squares
because 20 is not a
perfect square.
10. Yes
1 - 9​y ​4​
2
​1 ​2​- ​(3​y 2​ ​) ​ ​
2
(1 + 3​y ​ ​)(1 - 3​y 2​ ​)
12. No, (16​x ​2​+ 49) is not a
11. Yes
2
6
difference of 2 squares
25​m ​ ​- 4​m ​ ​
2
because the last term is
​(5m) ​2​- ​(2​m 3​ ​) ​ ​
2
2
2
not negative.
​m ​ ​(5 + 2​m ​ ​)(5 - 2​m ​ ​)
248 Holt McDougal Algebra 1
27. Let ℓ be the length.
​ℓ 2​ ​- 36
​ℓ 2​ ​- ​6 ​2​
(ℓ + 6)(ℓ - 6)
28. Let a be Michael’s age.
​a ​2​- 8a + 16
​a 2​ ​- 2(a)(4) + ​4 ​2​
(a -​ 4) ​2​
4a. 36​d ​ ​- 36d + 9
1
​(6d) ​2​- 2(6d)(3) + 3
​  ​2​
2
​(6d - 3) ​ ​
ℓ = (6d - 3) in
29. Let v be the speed.
2
2​v ​ ​+ 2v - 12
2(​v 2​ ​+ v - 6)
2(v + 3)(v - 2)
30. Let h be Jessie’s height.
3​h ​3​+ 3​h 2​ ​- 6h
3h(​h ​2​+ h - 2)
3h(h + 2)(h - 1)
b. The perimeter of the square is 4(6d - 3) in.
31. A = (9x)(8x) - (8y)(4y)
= 72​x ​2​- 32​y 2​ ​
= 8(9​x ​2​- 4​y 2​ ​)
= 8(3x + 2y)(3x - 2y)
13. Yes
​r 4​ ​- ​t 2​ ​
2
​(​r ​2​) ​ ​- ​t 2​ ​
2
(​r ​ ​+ t)(​r 2​ ​- t)
2
c. d = 2
6d - 3 = 6(2) - 3 = 9
4(6d - 3) = 4(9) = 36
​(6d - 3) ​2​= ​(9) ​2​= 81
When d = 2 in, the length of a side is 9 in, the
perimeter is 36 in, and the area is 81 i​n ​2​.
2
15. Yes, 5(​x ​ ​+ 3x + 1) is completely factored.
16. No
6x(5​x 2​ ​- x)
6x[5x(x) - 1(x)]
6​x ​2​(5x - 1)
17. No
3t(​t 4​ ​- 9)
2
3t[​(​t 2​ ​) ​ ​- ​3 ​2​]
2
3t(​t ​ ​+ 3)(​t 2​ ​- 3)
18. No.
2(​m ​2​- 10m + 25)
2[​m ​2​- 2(m)(5) + 5
​  ​2​]
2
2​(m - 5) ​ ​
Study guide: review
1. Prime factorization
2. Greatest common factor
7-1 FACTORS AND GREATEST
COMMON FACTORS
3. 12 = 2 · 2 · 3 = 2
​  ​2​· 3
4. 20 = 2 · 2 · 5 = ​2 ​2​· 5
5. 32 = 2 · 2 · 2 · 2 · 2 = ​2 ​5​ 6. 23 is a prime number
7. 40 = 2 · 2 · 2 · 5 = 2
​  ​3​· 5
6
8. 64 = 2 · 2 · 2 · 2 · 2 · 2 = 2
​  ​ ​
19. Yes, 3(2​y ​ ​- 5)(y + 1) is completely factored.
9. 66 = 2 · 3 · 11
20. No
(2n + 6)(n - 4)
2(n + 3)(n - 4)
11. 15 = 3 · 5
50 = 2 · 2 · 5
The GCF of 15 and 50 is 5.
2
3
2
21. 3​x ​ ​- 12​x ​ ​+ 12x
3x(​x 2​ ​- 4x + 4)
3x[(​x ​2​- 2(x)(2) + 2
​  ​2​]
2
3x(x - ​2) ​ ​
3
22. 16​m ​ ​- 4m
4m(4​m 2​ ​- 1)
4m[(2m​) ​2​- ​1 ​2​]
4m(2m + 1)(2m - 1)
3
23. 5​x ​ ​y - 45xy
5xy(​x 2​ ​- 9)
5xy(​x ​2​- ​3 ​2​)
5xy(x + 3)(x - 3)
2
24. 3​t ​ ​+ 5t - 1
​_
Factors
of 3 Factors
    
of -1 Outer + Inner
___________________________________
​1 and 3 1 and -1 1(-1) + 3(1) = -2 7
1 and 3 -1 and 1 1(1) + 3(-1) = 2 7
3​t ​2​+ 5t - 1 cannot be factored.
2
25.​3c ​ ​+ 12c - 63
3(​c 2​ ​+ 4c - 21)
​Factors
of -21  
Sum
__________________
​-1 and 21 20 7
-3 and 7 4 3
3(c - 3)(c + 7)
26.​x ​5​- 81x
x(​x 4​ ​- 81)
2
x[​(​x ​2​) ​ ​- ​9 ​2​]
2
x(​x ​ ​+ 9)(​x 2​ ​- 9)
x(​x ​2​+ 9)(x + 3)(x - 3)
10. 114 = 2 · 3 · 19
12. 36 = 2 · 2 · 3 · 3
132 = 2 · 2 · 3 · 11
The GCF of 36 and 132 is 12.
13. 29 is a prime number.
30 = 2 · 3 · 5
The GCF of 29 and 30 is 1.
14. 54 = 2 · 3 · 3 · 3
81 = 3 · 3 · 3 · 3
The GCF of 54 and 81 is 27.
15. 20 = 2 · 2 · 5
48 = 2 · 2 · 2 · 2 · 3
The GCF of 20 and 48 is 4.
16. 9m = 3 · 3 · m
3 is a prime number.
The GCF of 9m and 3 is 3.
17. 4x = 2 · 2 · x
2​x ​2​= 2 · x · x
The GCF of 4x and 2​x 2​ ​is 2x.
4
18. -18​b ​ ​= -1 · 2 · 3 · 3 · b · b · b · b
27​b ​2​= 3 · 3 · 3 · b · b
The GCF of -18​b 4​ ​and 27​b 2​ ​is 9​b 2​ ​.
19. 100r = 2 · 2 · 5 · 5 · r
5
25​r ​ ​= 5 · 5 · r · r · r · r · r
The GCF of 100r and 2
​ 5r 5​ ​is 25r.
249 Holt McDougal Algebra 1
20. The 42 types of boxed nails and 36 types of boxed
screws must be divided into groups of equal size.
The number of boxes in each row must be a
common factor of 42 and 36.
factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
The GCF of 42 and 36 is 6.
The greatest possible number of boxes in each row
is 6. Find the number of rows.
42 types of boxed nails
​ ___________________
   
  
 ​= 7 rows
6 boxes per row
36 types of boxed screws
   
  
 ​= 6 rows
​ _____________________
6 boxes per row
When the greatest possible number of boxes is in
each row, there are 13 rows in total.
7-2 Factoring by GCF
21. 5x = 5 · x
15​x 3​ ​= 3 · 5 · x · x · x
GCF: 5 · x = 5x
5x - 15​x 3​ ​= 5x(1) - 5x(3​x 2​ ​)
= 5x(1 - 3​x ​2​)
22. 16b = 2 · 2 · 2 · 2 · b
32 = 2 · 2 · 2 · 2 · 2
GCF: 2 · 2 · 2 · 2 = 16
-16b + 32 = 16(-b) + 16(2)
= 16(-b + 2)
23. -1(14v + 21)
14v = 2 · 7 · v
21 = 3 · 7
GCF: 7
-1(14v + 21)
-1[7(2v) + 7(3)]
-7(2v + 3)
24. 4​a ​2​= 2 · 2 · a · a
12a = 2 · 2 · 3 · a
8 =2·2·2
GCF: 2 · 2 = 4
4​a 2​ ​- 12a - 8 = 4(​a 2​ ​) - 4(3a) - 4(2)
= 4(​a 2​ ​- 3a - 2)
5
25. 5​g ​ ​ = 5 · g · g · g · g · g
10​g 3​ ​= 2 · 5 · g · g · g
15g = 3 · 5 · g
GCF: 5 · g = 5g
5​g 5​ ​- 10​g 3​ ​- 15g = 5g(​g 4​ ​) - 5g(2​g 2​ ​) - 5g(3)
= 5g(​g ​4​- 2​g 2​ ​- 3)
= 5g(​g ​2​- 3)(​g 2​ ​+ 1)
2
26. 40​p ​ ​= 2 · 2 · 2 · 5 · p · p
10p = 2 · 5 · p
30 = 2 · 3 · 5
GCF: 2 · 5 = 10
40​p 2​ ​- 10p + 30 = 10(4​p 2​ ​) - 10(p) + 10(3)
= 10(4​p 2​ ​- p + 3)
2
27. 6​x ​ ​= 2 · 3 · x · x
5x = 5 · x
GCF: x
6​x 2​ ​+ 5x = x(6x) + x(5)
= x(6x + 5)
The dimension of the lot is (6x + 5) ft by x ft.
28. 2x(x - 4) + 9(x - 4)
(2x + 9)(x - 4)
29. t(3t + 5) - 6(3t + 5)
(t - 6)(3t + 5)
30. 5(6 - n) - 3n(6 - n)
(5 - 3n)(6 - n)
31. b(b + 4) + 2(b + 4)
(b + 2)(b + 4)
32.​x 2​ ​(x - 3) + 7(x - 3)
(​x 2​ ​+ 7)(x - 3)
33.​n ​ ​+ n - 4​n ​ ​- 4
(​n ​3​- 4​n 2​ ​) + (n - 4)
​n ​2​(n - 4) + 1(n - 4)
(​n 2​ ​+ 1)(n - 4)
3
2
2
34. 6​b ​ ​- 8b + 15b - 20
(6​b 2​ ​- 8b) + (15b - 20)
2b(3b - 4) + 5(3b - 4)
(2b + 5)(3b - 4)
35. 2​h 3​ ​- 7h + 14​h 2​ ​- 49
(2​h ​3​+ 14​h 2​ ​) - (7h + 49)
2​h ​2​(h + 7) - 7(h + 7)
(2​h 2​ ​- 7)(h + 7)
2
3
2
37. 10​m ​ ​+ 15​m ​ ​- 2m - 3
36. 3​t ​ ​+ 18t + t + 6
(3​t 2​ ​+ 18t) + (t + 6)
(10​m ​3​+ 15​m 2​ ​) - (2m + 3)
3t(t + 6) + 1(t + 6)
5​m ​2​(2m + 3) - 1(2m + 3)
(3t + 1)(t + 6)
(5​m 2​ ​- 1)(2m + 3)
2
38. 8​p 3​ ​+ 4p - 6​p 2​ ​- 3
39. 5r - 10 + 2r - r​  ​ ​
3
2
2
(8​p ​ ​- 6​p ​ ​) + (4p - 3)
(5r - 10) - (​r ​ ​- 2r)
5(r - 2) - r(r - 2)
2​p ​2​(4p - 3) + 1(4p - 3)
2
-1(r - 5)(r - 2)
(2​p ​ ​+ 1)(4p - 3)
3
2
40.​b 3​ ​- 5b + 15 - 3​b 2​ ​
41. 6t - ​t ​ ​- 4​t ​ ​+ 24
3
2
3
2
-(​
t
 
​
​
+
4​
t
 
​
)
​
+
(6t + 24)
(​b ​ ​- 3​b ​ ​) - (5b - 15)
2
2
-​t ​ ​(t + 4) + 6(t + 4)
​b ​ ​(b - 3) - 5(b - 3)
-1(​t ​2​- 6)(t + 4)
(​b 2​ ​- 5)(b - 3)
2
3. d - ​d 2​ ​+ d - 1
4
42. 12h - 3​h ​ ​+ h - 4
2
-1(3​h ​ ​- 12h) + (h - 4)
-(​d ​2​- d) + (d - 1)
-3h(h - 4) + 1(h - 4)
-d(d - 1) + 1(d - 1)
-1(3h - 1)(h - 4)
-1(d - 1
​ ) ​2​
2
2
45. 5t - ​t ​ ​- t + 5
44. 6b - 5​b ​ ​+ 10b - 12
-(5​b 2​ ​- 6b) + (10b - 12)
-(​t 2​ ​- 5t) - (t - 5)
-b(5b - 6) + 2(5b - 6)
-t(t - 5) - 1(t - 5)
-1(b - 2)(5b - 6)
-1(t + 1)(t - 5)
2
46. 8​b ​2​- 2​b 3​ ​- 5b + 20
47. 3r - 3​r ​ ​- 1 + r
3
2
2
-(2​b ​ ​- 8​b ​ ​) - (5b - 20)
-(3​r ​ ​- 3r) + (r - 1)
-3r(r - 1) + 1(r - 1)
-2​b ​2​(b - 4) - 5(b - 4)
2
-1(3r - 1)(r - 1)
-1(2​b ​ ​+ 5)(b - 4)
48. Left rectangle: x(2x + 3) = 2​x ​2​+ 3x
Right rectangle: 2(4x + 6) = 8x + 12
Combined: (2​x ​2​+ 3x) + (8x + 12)
= (2​x ​2​+ 8x) + (3x + 12)
= 2x(x + 4) + 3(x + 4)
= (2x + 3)(x + 4)
7-3 factoring ​x ​2​+ bx + c
49.​x 2​ ​+ 6x + 5
(x + 1)(x + 5)
2
50.​x ​ ​+ 6x + 8
(x + 2)(x + 4)
2
51.​x ​ ​+ 8x + 15
(x + 3)(x + 5)
250 Holt McDougal Algebra 1
52.​x 2​ ​- 8x + 12
(x - 2)(x - 6)
2
53.​x ​ ​+ 10x + 25
​x ​2​+ 2(x)(5) + 5
​  ​2​
2
​(x + 5) ​ ​
2
54.​x ​ ​- 13x + 22
(x - 2)(x - 11)
2
55.​x ​ ​+ 24x + 80
(x + 4)(x + 20)
2
56.​x ​ ​- 26x + 120
(x - 6)(x - 20)
2
57.​x ​ ​+ 5x - 84
(x + 12)(x - 7)
2
58.​x ​ ​- 5x - 24
(x + 3)(x - 8)
2
59.​x ​ ​- 3x - 28
(x + 4)(x - 7)
2
60.​x ​ ​+ 4x - 5
(x + 5)(x - 1)
2
61.​x ​ ​+ x - 6
(x + 3)(x - 2)
2
62.​x ​ ​+ x - 20
(x + 5)(x - 4)
2
63.​x ​ ​- 2x - 48
(x + 6)(x - 8)
2
64.​x ​ ​- 5x - 36
(x + 4)(x - 9)
2
65.​x ​ ​- 6x - 72
(x + 6)(x - 12)
2
66.​x ​ ​- 3x - 70
(x + 7)(x - 10)
2
67.​x ​ ​+ 14x - 120
(x + 20)(x - 6)
2
68.​x ​ ​+ 6x - 7
(x + 7)(x - 1)
2
69.​y ​ ​+ 8y + 15
(y + 3)(y + 5)
ℓ= (y + 5) m
w = (y + 3) m
7-4 factoring ​ax ​2​+ bx + c
70. 2​x 2​ ​+ 11x + 5
​___________________________________
Factors
of 2 Factors
    
of 5 Outer + Inner
​1 and 2 1 and 5 1(5) + 2(1) = 7
1 and 2 5 and 1 1(1) + 2(5) = 11
(x + 5)(2x + 1)
71. 3​x 2​ ​+ 10x + 7
​___________________________________
Factors
of 3 Factors
    
of 7 Outer + Inner​
1 and 3 1 and 7 1(7) + 3(1) = 10
(x + 1)(3x + 7)
2
72. 2​x ​ ​- 3x + 1
​___________________________________
Factors
of 2 Factors
    
of 1 Outer + Inner​
1 and 2 -1 and -1 1(-1) + 2(-1) = -3
(x - 1)(2x - 1)
73. 3​x 2​ ​+ 8x + 4
​___________________________________
Factors
of 3 Factors
    
of 4 Outer + Inner
​1 and 3 1 and 4 1(4) + 3(1) = 7
1 and 3 4 and 1 1(1) + 3(4) = 13
1 and 3 2 and 2 1(2) + 3(2) = 8
(x + 2)(3x + 2)
74. 5​x 2​ ​+ 28x + 15
​____________________________________
Factors
of 5 Factors
    
of 15 Outer + Inner
​1 and 5 1 and 15 1(15) + 5(1) = 20
1 and 5 15 and 1 1(1) + 5(15) = 76
1 and 5 3 and 5 1(5) + 5(3) = 20
1 and 5 5 and 3 1(3) + 5(5) = 28
(x + 5)(5x + 3)
75. 6​x 2​ ​- 19x + 15
​____________________________________
Factors
of 6 Factors
    
of 15 Outer + Inner​
1 and 6 -1 and -15 1(-15) + 6(-1) = -21
1 and 6 -15 and -1 1(-1) + 6(-15) = -91
1 and 6
-3 and -5 1(-5) + 6(-3) = -23
1 and 6 -5 and -3 1(-3) + 6(-5) = -33
2 and 3 -1 and -15 2(-15) + 3(-1) = -33
2 and 3 -15 and -1 2(-1) + 3(-15) = -47
2 and 3 -3 and -5 2(-5) + 3(-3) = -19
(2x - 3)(3x - 5)
76. 4​x ​2​+ 13x + 10
​____________________________________
Factors
of 4 Factors
    
of 10 Outer + Inner
​1 and 4 1 and 10 1(10) + 4(1) = 14
1 and 4 10 and 1 1(1) + 4(10) = 41
1 and 4 2 and 5 1(5) + 4(2) = 13
(x + 2)(4x + 5)
77. 3​x ​2​+ 10x + 8
​___________________________________
Factors
of 3 Factors
    
of 8 Outer + Inner
​1 and 3 1 and 8 1(8) + 3(1) = 11
1 and 3 8 and 1 1(1) + 3(8) = 25
1 and 3 2 and 4 1(4) + 3(2) = 10
(x + 2)(3x + 4)
78. 7​x 2​ ​- 37x + 10
​____________________________________
Factors
of 7 Factors
    
of 10 Outer + Inner​
1 and 7 -1 and -10 1(-10) + 7(-1) = -17
1 and 7 -10 and -1 1(-1) + 7(-10) = -71
1 and 7 -2 and -5 1(-5) + 7(-2) = -19
1 and 7 -5 and -2 1(-2) + 7(-5) = -37
(x - 5)(7x - 2)
79. 9​x ​2​+ 18x + 8
​___________________________________
Factors
of 9 Factors
    
of 8 Outer + Inner​
1 and 9 1 and 8 1(8) + 9(1) = 17
1 and 9 8 and 1 1(1) + 9(8) = 73
1 and 9 2 and 4 1(4) + 9(2) = 22
1 and 9 4 and 2 1(2) + 9(4) = 38
3 and 3 1 and 8 3(8) + 3(1) = 27
3 and 3 2 and 4 3(4) + 3(2) = 18
(3x + 2)(3x + 4)
80. 2​x 2​ ​- x - 1
​_
Factors
of 2 Factors
    
of -1 Outer + Inner
___________________________________
​1 and 2 1 and -1 1(-1) + 2(1) = 1
1 and 2 -1 and 1 1(1) + 2(-1) = -1
(x - 1)(2x + 1)
251 Holt McDougal Algebra 1
81. 3​x 2​ ​- 11x - 4
​Factors
of 3 Factors
    
of -4 Outer + Inner​
_
___________________________________
1 and 3 1 and -4 1(-4) + 3(1) = -1
1 and 3 -1 and 4 1(4) + 3(-1) = 1
1 and 3 2 and -2 1(-2) + 3(2) = 4
1 and 3 -2 and 2 1(2) + 3(-2) = -4
1 and 3 4 and -1 1(-1) + 3(4) = 11
1 and 3 -4 and 1 1(1) + 3(-4) = -11
(x - 4)(3x + 1)
88. -4​x ​2​+ 8x + 5
-1(4​x 2​ ​- 8x - 5)
​_
Factors
of 4 Factors
    
of -5 Outer + Inner
___________________________________
​1 and 4 1 and -5 1(-5) + 4(1) = -1
1 and 4 -1 and 5 1(5) + 4(-1) = 1
1 and 4 5 and -1 1(-1) + 4(5) = 19
1 and 4 -5 and 1 1(1) + 4(-5) = -19
2 and 2 1 and -5 2(-5) + 2(1) = -8
-1(2x + 1)(2x - 5)
82. 2​x ​2​- 11x + 5
​ __________________________________
Factors
of 2 Factors
    
of 5 Outer + Inner
_
​1 and 2
-1 and -5 1(-5) + 2(-1) = -7
1 and 2 -5 and -1 1(-1) + 2(-5) = -11
(x - 5)(2x - 1)
89. -10​x ​2​+ 11x + 6
-1(10​x 2​ ​- 11x - 6)
​_____________________________________
Factors
of 10     
Factors of -6 Outer + Inner
​1 and 10 1 and -6 1(-6) + 10(1) = 4
1 and 10 -1 and 6 1(6) + 10(-1) = -4
1 and 10 2 and -3 1(-3) + 10(2) = 17
1 and 10 -2 and 3 1(3) + 10(-2) = -17
1 and 10 3 and -2 1(-2) + 10(3) = 28
1 and 10 -3 and 2 1(2) + 10(-3) = -28
1 and 10 6 and -1 1(-1) + 10(6) = 59
1 and 10 -6 and 1 1(1) + 10(-6) = -59
2 and 5 1 and -6 2(-6) + 5(1) = -7
2 and 5 -1 and 6 2(6) + 5(-1) = 7
2 and 5 2 and -3 2(-3) + 5(2) = 4
2 and 5 -2 and 3 2(3) + 5(-2) = -4
2 and 5 3 and -2 2(-2) + 5(3) = 11
2 and 5 -3 and 2 2(2) + 5(-3) = -11
-1(2x - 3)(5x + 2)
83. 7​x 2​ ​- 19x - 6
​Factors
of 7 Factors
    
of -6 Outer + Inner
_
___________________________________
​1 and 7 1 and -6 1(-6) + 7(1) = 1
1 and 7 -1 and 6 1(6) + 7(-1) = -1
1 and 7 2 and -3 1(-3) + 7(2) = 11
1 and 7 -2 and 3 1(3) + 7(-2) = -11
1 and 7 3 and -2 1(-2) + 7(3) = 19
1 and 7 -3 and 2 1(2) + 7(-3) = -19
(x - 3)(7x + 2)
84. 5​x ​2​- 9x - 2
​Factors
of 5 Factors
    
of -2 Outer + Inner​
_
___________________________________
1 and 5 1 and -2 1(-2) + 5(1) = 3
1 and 5 -1 and 2 1(2) + 5(-1) = -3
1 and 5 2 and -1 1(-1) + 5(2) = 9
1 and 5 -2 and 1 1(1) + 5(-2) = -9
(x - 2)(5x + 1)
90. 12​x ​2​+ 4x - 15x - 5
(12​x 2​ ​+ 4x) - (15x + 5)
4x(3x + 1) - 5(3x + 1)
(4x - 5)(3x + 1)
85. -6​x ​2​- x + 2
-1(6​x 2​ ​+ x - 2)
​_
Factors
of 6 Factors
    
of -2 Outer + Inner
___________________________________
​1 and 6 1 and -2 1(-2) + 6(1) = 4
1 and 6 -1 and 2 1(2) + 6(-1) = -4
1 and 6 2 and -1 1(-1) + 6(2) = 11
1 and 6 -2 and 1 1(1) + 6(-2) = -11
2 and 3 1 and -2 2(-2) + 3(1) = -1
2 and 3 -1 and 2 2(2) + 3(-1) = 1
-1(2x - 1)(3x + 2)
7-5 FACTORING SPECIAL PRODUCTS
91. Yes
​x 2​ ​+ 12x + 36
​x 2​ ​+ 2(x)(6) + 6
​  ​2​
(x​+ 6) ​2​
2
92. No, (​x ​ ​+ 5x + 25) is not a perfect-square trinomial
because 5x ≠ 2(x)(5).
86. 6​x ​2​- x - 5
​Factors
of 6 Factors
    
of -5 Outer + Inner
_
___________________________________
​1 and 6 1 and -5 1(-5) + 6(1) = 1
1 and 6 -1 and 5 1(5) + 6(-1) = -1
(x - 1)(6x + 5)
2
93. No, (4​x ​ ​- 2x + 1) is not a perfect-square trinomial
because -2x ≠ 2(2x)(1).
87. 6​x ​2​+ 17x - 14
​ ____________________________________
Factors
of 6      
Factors of -14 Outer + Inner
_
​1 and 6 1 and -14 1(-14) + 6(1) = -8
1 and 6 -1 and 14 1(14) + 6(-1) = 8
1 and 6 2 and -7 1(-7) + 6(2) = 5
1 and 6 -2 and 7 1(7) + 6(-2) = -5
1 and 6 7 and -2 1(-2) + 6(7) = 40
1 and 6 -7 and 2 1(2) + 6(-7) = -40
1 and 6 14 and -1 1(-1) + 6(14) = 83
1 and 6 -14 and 1 1(1) + 6(-14) = -83
2 and 3 1 and -14 2(-14) + 3(1) = -25
2 and 3 -1 and 14 2(14) + 3(-1) = 25
2 and 3 2 and -7 2(-7) + 3(2) = -8
2 and 3 -2 and 7 2(7) + 3(-2) = 8
2 and 3 7 and -2 2(-2) + 3(7) = 17
(2x + 7)(3x - 2)
94. Yes
9​x ​2​+ 12x + 4
(3​x) ​2​+ 2(3x)(2) + 2
​  ​2​
(3x +​ 2) ​2​
2
95. No, (16​x ​ ​+ 8x + 4) is not a perfect-square trinomial
because 8x ≠ 2(4x)(2).
96. Yes
​x ​2​+ 14x + 49
​x 2​ ​+ 2(x)(7) + 7
​  ​2​
2
(x + 7
​ ) ​ ​
97. Yes
100​x ​2​- 81
(10​x) ​2​- ​9 ​2​
(10x + 9)(10x - 9)
98. No, (​x ​2​- 2) is not a difference of 2 squares
because 2 is not a perfect square.
4
6
9. No, (5​x ​ ​- 10​y ​ ​) is not a difference of 2 squares
9
because 5​x ​4​- 10​y 6​ ​= 5(​x 4​ ​- 2​y 6​ ​) and 2​y 6​ ​is not a
perfect square.
252 Holt McDougal Algebra 1
101. No, (121​b ​2​+ 9​c 8​ ​) is
100. Yes
2
3 2
(-1​2) ​ ​- (​x ​ ​​) ​
not a difference of
2 squares because
​
(-12 + ​x ​3​)(-12 - ​x 3​ ​)
3
3
the operation between
-1(​x ​ ​+ 12)(​x ​ ​- 12)
the terms is addition,
and not subtracting.
102. Yes
100​p ​2​- 25​q 2​ ​
25(4​p ​2​- ​q 2​ ​)
25[(2p​) ​2​- ​q 2​ ​]
25(2p + q)(2p - q)
03. difference of 2 squares
1
​x ​2​- 25
​x ​2​- ​5 ​2​
(x + 5)(x - 5)
04. Perfect-square trinomial
1
​x ​2​+ 20x + 100
​x ​2​+ 2(x)(10) + 1
​ 0 ​2​
(x + 10​) ​2​
05. Difference of 2 squares
1
2
4
​j ​ ​- ​k ​ ​
2
​j ​ ​- (​k 2​ ​​) ​2​
(j + ​k ​2​)(j - k​  2​ ​)
06. Perfect-square trinomial
1
2
9​x ​ ​- 42x + 49
(3x​) ​2​- 2(3x)(7) + 7
​  ​2​
(3x - 7​) ​2​
07. Perfect-square trinomial
1
2
81​x ​ ​+ 144x + 64
(9x​) ​2​+ 2(9x)(8) + 8
​  ​2​
2
(9x + ​8) ​ ​
08. Difference of 2 squares
1
4
6
16​b ​ ​- 121​c ​ ​
(4​b ​2​​) ​2​- (11​c 3​ ​​) ​2​
(4​b ​2​+ 11​c ​3​)(4​b 2​ ​- 11​c ​3​)
7-6 CHOOSING A FACTORING METHOD
09. No
1
4​x ​2​+ 10x + 6
(4x + 6)(x + 1)
2(2x + 3)(x + 1)
111. No
4
​b ​ ​- 81
(​b 2​ ​+ 9)(​b 2​ ​- 9)
(​b ​2​+ 9)(b + 3)(b - 3)
2
112. Yes, (x - 3​) ​ ​is completely factored.
2
113. 4​x ​ ​- 64
4(​x 2​ ​- 16)
4(x + 4)(x - 4)
4
3
114. 3​b ​ ​- 6​b ​ ​- 24​b ​ ​
3​b ​3​(​b 2​ ​- 2b - 8)
3​b ​2​(b + 2)(b - 4)
4 3
2 5
115.​a ​ ​​b ​ ​- ​a ​ ​​b ​ ​
​a ​2​​b 3​ ​(​a 2​ ​- ​b 2​ ​)
​a ​2​​b 3​ ​(a + b)(a - b)
2
117. 5​x ​ ​+ 20x + 15
5(​x 2​ ​+ 4x + 3)
5(x + 1)(x + 3)
4
2
118. 2​x ​ ​- 50​x ​ ​
2​x ​2​(​x 2​ ​- 25)
2​x ​2​(x + 5)(x - 5)
119. 8t + 32 + 2st + 8s
2(st + 4s + 4t + 16
2[(st + 4s) + (4t + 16)]
2[s(t + 4) + 4(t + 4)]
2(s + 4)(t + 4)
20. 25​m 3​ ​- 90​m 2​ ​- 40m
1
5m(5​m ​2​- 18m - 8)
​_
Factors
of 5 Factors
    
of -8 Outer + Inner​
___________________________________
1 and 5 1 and -8 1(-8) + 5(1) = -3
1 and 5 -1 and 8 1(8) + 5(-1) = 3
1 and 5 2 and -4 1(-4) + 5(2) = 6
1 and 5 -2 and 4 1(4) + 5(-2) = -6
1 and 5 4 and -2 1(-2) + 5(4) = 18
1 and 5 -4 and 2 1(2) + 5(-4) = -18
5m(m - 4)(5m + 2)
21. 32​x ​4​- 48​x 3​ ​+ 8​x 2​ ​- 12x
1
4x(8​x ​3​- 12​x 2​ ​+ 2x - 3)
4x[(8​x ​3​- 12​x 2​ ​) + (2x - 3)]
4x[4​x ​2​(2x - 3) + 1(2x - 3)]
4x(4​x 2​ ​+ 1)(2x - 3)
4
3 2
2 3
22. 6​s ​ ​t + 12​s ​ ​​t ​ ​+ 6​s ​ ​​t ​ ​
1
6​s ​2​t(​s 2​ ​+ 2st + t​ 2​ ​)
6​s ​2​t(s + t​) ​2​
3
110. Yes, 3(​y ​2​+ 25) is completely factored.
5
116.​t ​20​- ​t 4​ ​
​t ​4​(​t 16
​ ​- 1)
​t ​4​(​t 8​ ​+ 1)(​t 8​ ​- 1)
​t ​4​(​t 8​ ​+ 1)(​t 4​ ​+ 1)(​t 4​ ​- 1)
​t ​4​(​t 8​ ​+ 1)(​t 4​ ​+ 1)(​t 2​ ​+ 1)(​t 2​ ​- 1)
​t ​4​(​t 8​ ​+ 1)(​t 4​ ​+ 1)(​t 2​ ​+ 1)(t + 1)(t - 1)
2
23. 10​m ​ ​+ 4​m ​ ​- 90m - 36
1
2(5​m ​3​+ 2​m 2​ ​- 45m - 18)
2[(5​m ​3​+ 2​m 2​ ​) - (45m + 18)]
2[​m ​2​(5m + 2) - 9(5m + 2)]
2(​m 2​ ​- 9)(5m + 2)
2(m + 3)(m - 3)(5m + 2)
chapter test
1. 3​t 4​ ​= 3 · t · t · t · t
8​t ​2​= 2 · 2 · 2 · t · t
The GCF of 3​t 2​ ​and 8
​ t ​2​is ​t 2​ ​.
3
2. 2​y ​ ​ = 2 · y · y · y
-12y = -1 · 2 · 2 · 3 · y
The GCF of 2
​ y 3​ ​and -12y is 2y.
5
3. 15​n ​ ​= 3 · 5 · n · n · n · n · n
9​n 4​ ​ = 3 · 3 · n · n · n · n
The GCF of 15​n 5​ ​and 9​n 4​ ​is 3​n 4​ ​.
4. 360 = 2 · 2 · 2 · 3 · 3 · 5 = 2
​  ​3 ​· ​3 ​2​· 5
253 Holt McDougal Algebra 1
5. The 16 Liberty nickels, 24 Buffalo nuckels, and
40 Jefferson nickels must be divided into groups
of equal size. The number of nickels in each row
must be a common factor of 16, 24 and 40.
factors of 16: 1, 2, 4, 8, 16
factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
The GCF of 16, 24 and 40 is 8.
The greatest possible number of nickels in each row
is 8. Find the number of rows.
16 Liberty nickels
​ _______________
  
  
 ​= 2 rows
8 per row
19. Yes
​a ​2​+ 14a + 49
​a 2​ ​+ 2(a)(7) + 7
​  ​2​
2
(a + 7
​ ) ​ ​
2
20. No, (2​x ​ ​+ 10x + 25) is not a perfect-square
trinomial because 2​x 2​ ​is not a perfect square.
21. Yes
9​t ​2​- 6t + 1
(3t​) ​2​- 3(3t)(1) + 1
​  ​2​
2
(3t - 1
​ ) ​ ​
22. Yes
​b ​2​- 16
​b 2​ ​- ​4 ​2​
(b + 4)(b - 4)
24 Buffalo nickels
  
 ​
   = 3 rows
​ _______________
8 per row
23. No, (25​y 2​ ​- 10) is not a difference of 2 squares
because 10 is not a perfect square.
40 Jefferson nickels
​ _________________
  
 ​
   = 5 rows
8 per row
When the greatest possible number of nickels is in
each row, there are 10 rows in total.
24. Yes
9​a ​2​- ​b 10
​ ​
(3a​) ​2​- (​b 5​ ​​) ​2​
(3a + b
​  ​5​)(3a - b
​  5​ ​)
6. 24​m 2​ ​+ 4​m 3​ ​
4​m ​2​(6) + 4​m 2​ ​(m)
4​m ​2​(6 + m)
7. 9​x ​5​- 12x
3x(3​x 4​ ​) - 3x(4)
3x(3​x 4​ ​- 4)
8. -2​r 4​ ​- 6
-2(​r 4​ ​) - 2(3)
-2(​r 4​ ​+ 3)
9. 3(c - 5) + 4c(c - 5)
(3 + 4c)(c - 5)
2
25. 9​x ​ ​+ 30x + 25
(3x​) ​2​+ 2(3x)(5) + 5
​  ​2​
2
(3x + ​5) ​ ​
P = 4(3x + 5) ft
x = 4, P = 4(3x + 5) = 4[3(4) + 5] = 68
The perimeter is 68 ft when x = 4 ft.
26. No
(6x - 3)(x + 5)
3(2x - 1)(x + 5)
10. 10​x ​3​+ 4x - 25​x 2​ ​- 10
(10​x ​3​- 25​x 2​ ​) + (4x - 10)
5​x ​2​(2x - 5) + 2(2x - 5)
(5​x 2​ ​+ 2)(2x - 5)
3
27. Yes, (​v ​5​+ 10)(​v 5​ ​- 10) is completely factored.
28. Yes, (2b + 3)(3b - 2) is completely factored.
2
2
11. 4​y ​ ​- 4​y ​ ​- 3 + 3y
12. -5​t ​ ​+ 50t + 5
(4​y ​3​- 4​y 2​ ​) + (3y - 3)
-1(5​t 2​ ​- 50t - 5)
4​y ​2​(y - 1) + 3(y - 1)
-1[5(​t 2​ ​) - 5(10t) - 5(1)]
(4​y 2​ ​+ 3)(y - 1)
-5(​t ​2​- 10t - 1)
2
13.​x ​ ​+ 6x + 5
(x + 1)(x + 5)
2
14.​x ​ ​- 4x - 21
(x + 3)(x - 7)
2
15.​x ​ ​- 8x + 15
(x - 3)(x - 5)
16. 2​x 2​ ​+ 9x + 7
​ __________________________________
Factors
of 2 Factors
    
of 7 Outer + Inner​
_
1 and 2 1 and 7 1(7) + 2(1) = 9
(x + 1)(2x + 7)
2
3
2
29. 8​x ​ ​+ 72​x ​ ​+ 160x
8x(​x 2​ ​+ 9x + 20)
8x(x + 4)(x + 5)
5
3
30. 3​x ​ ​- 27​x ​ ​
3​x ​3​(​x 2​ ​- 9)
3​x ​3​(x + 3)(x - 3)
31. 8​x ​3​+ 64​x 2​ ​- 20x - 160
4(2​x ​3​+ 16​x 2​ ​- 5x - 40)
4[(2​x ​3​+ 16​x 2​ ​) - (5x + 40)]
4[2​x ​2​(x + 8) - 5(x + 8)]
4(2​x 2​ ​- 5)(x + 8)
4
7 6
32. c​d ​ ​- ​c ​ ​​d ​ ​
c​d ​4​(1 - c​  6​ ​​d 2​ ​)
c​d ​4​(1 + c​  3​ ​d)(1 - c​  3​ ​d)
2
17. 2​x ​ ​+ 9x - 18
​ ____________________________________
Factors
of 2      
Factors of -18 Outer + Inner
_
​1 and 2 1 and -18 1(-18) + 2(1) = -16
1 and 2 -1 and 18 1(18) + 2(-1) = 16
1 and 2 2 and -9 1(-9) + 2(2) = -5
1 and 2 -2 and 9 1(9) + 2(-2) = 5
1 and 2 3 and -6 1(-6) + 2(3) = 0
1 and 2 -3 and 6 1(6) + 2(-3) = 0
1 and 2 6 and -3 1(-3) + 2(6) = 9
(x + 6)(2x - 3)
33. 100​x ​ ​- 80x + 16
4(25​x 2​ ​- 20x + 4)
4[(5x​) ​2​- 2(5x)(2) + 2
​  ​2​]
4(5x - 2
​ ) ​2​
8
34. 7​m ​ ​- 7
7(​m 8​ ​- 1)
7(​m 4​ ​+ 1)(​m 4​ ​- 1)
7(​m ​4​+ 1)(​m 2​ ​+ 1)(​m 2​ ​- 1)
7(​m ​4​+ 1)(​m 2​ ​+ 1)(m + 1)(m - 1)
18. -3​x 2​ ​- 2x + 8
-1(3​x 2​ ​+ 2x - 8)
​_
Factors
of 3 Factors
    
of -8 Outer + Inner​
___________________________________
1 and 3 1 and -8 1(-8) + 3(1) = -5
1 and 3 -1 and 8 1(8) + 3(-1) = 5
1 and 3 2 and -4 1(-4) + 3(2) = 2
-1(x + 2)(3x - 4)
254 Holt McDougal Algebra 1