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SECTION 6.3
(e)
H-
(x + 2 )
3
+ /
Special Factoring
337
3
EXERCISE 4
=
Factor each polynomial.
(a) 1000 + z
(b)
81a
6
(c) (x-
3
[(.v + 2 ) +
=
(x
t][(x
2
+ 2 +
t){x
+ 2)
2
-
(x +
+ Ax + 4 -
xt
2
2)t
+ l]
Sum of cubes
2
-
2t + t )
Multiply.
NOW TRY *
3
+ 3b
3
3
3) + y
/$±CAUTION
3
A c o m m o n error w h e n factoring x
3
3
+ y
3
or x
—y
is to t h i n k that
the jry-term has a coefficient o f 2. Since there is n o coefficient o f 2, expressions o f
2
2
the f o r m x
+ xy + y
2
2
and x
— xy + y
usually cannot be factored further.
The special types o f factoring are summarized here. These should be
memorized.
3es of Factorin
DiflCr^or^uares
Perfect Sqtrar«4Vinoniial
NOW TRY ANSWERS
4. (a) (10 + z) •
x
-y
x
+ 2xy + y = (x + y)
x
— 2xy + y
2
= (x + y)(x - y)
2
2
2
2
2
lOz + z )
(100 -
2
2
(b) 3(3a + b) •
(9a - 3a 6 + b )
(c) (x - 3 + y) •
(x - 6x + 9 - xy +
3y + y )
4
2
2
3
KJf"
Sum of Cubes
—
(JC —
y)
2
x — v = (JC — v)(* + xy + v )
Difference of Cubes*?* C *
2
2
3
JC
3
+j =
3
2
(JC + J ) ( J C
—
2
2
jcp + y )
2
2
6.3
EXERCISES
WAKN
Concept Check
© Complete solution available
on the Video Resources on DVD
/ A " "
—
Work each problem.
1. Which o f the following binomials are differences o f squares?
2
2
A . 64 - k
B . 2x
2
- 25
C. k
+ 9
4
D. 4z -
49
2. Which o f the following binomials are sums or differences o f cubes?
3
6
A . 64 + r
3
B . 125 - p
C. 9 x + 125
3
D. (x + y)
- 1
3. Which o f the following trinomials are perfect squares?
2
A. x
- Sx 4
2
16
B . 4 m + 20m + 25
2
2
C. 9z + 30z + 25
D. 25p
- 4 5 / 7 + 81
4. O f the 12 polynomials listed in Exercises 1-3, which ones can be factored by the methods
o f this section?
2
5. The binomial 4 x + 64 is an example o f a sum o f two squares that can be factored. Under
what conditions can the sum o f two squares be factored?
6. Insert the correct signs in the blanks.
(a) 8 + m = (2 — m ) ( 4 — 2m —
3
(b)
3
n -
1 = (n —
2
1)(« — n —
Factor each polynomial. See Examples
©
2
7. p
-
2
16
8. k
2
10. 36m - 25
4
4
13. 64m - 4y
2
m)
1)
1-4.
-
2
9
2
11. 18a 4
14. 243x -
9. 25x - 4
98ft
3/
4
2
2
2
12. 32c -
98d
2
15. (y + z ) - 81
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represents a copyright violation.
338
CHAPTER 6
Factoring
2
16. {h + k)
4
19. p
2
- 9
17. 16 - {x + 3y)
4
- 256
2
22. x
20. a
+ Wx + 25
2
2
2
1
29. x
35. (a - b)
©37.
x
2
36. (m - n)
3
50. z
3
3
125p
3
56. 250x
6
+ 125
59. m -
9
lOOOx
3
2
1
48. 8 w -
3
125
3
3
51. 6 4 g -
3
3
21h
3
54. 512/ + 27s
3
3
+ \6y
57. {y + z ) + 64
125
60. 2 7 r + 1
62. 64 - 729/>
64. Concept Check
Consider (x — y)
x — 2xy + y — 25 correct?
+ 343
45. 8x + 1
53. 343p + 125a
3
3
3
3
3
52. 27a - 8b
58. (p - q)
-
+ 2kh + 4
+ 4(m - n) + 4
42. r
47. 125x - 216
3
2
s
2
+ 64
3
3
2
- h
39. 216 - i
44. 729 + x
3
12r + 9 -
- 64
3
55. 24n + 81p
2
2
+ 6{x + y) + 9
2
3
46. 2 7 y + . 1
61. 27 -
2
© 41. x
43. 1000 + y
3
-
30. - *
34. (x + y)
38. y
- 8y
1
2
32. 80z - 40zw + 5 w
- 27
3
27. 4 r
2
3
49. x
+ 2y -
+ 8(a - b) + 16
40. 512 - m
3
2
+ 2{p + q) + 1
2
3
2
- y
- 6k + 9
24. 9>> + 6^z + z
2
2
2t)
2
26. 25c - 20c + 4 - d
31. 98m + 84mn + 1 8 «
33. {p + q)
2
2
- 24a + 16 - b
2
21. k
© 23. 4z + 4zw + w
25. 16m - 8m + 1 - n
28. 9 a
2
- 625
2
2
18. 64 - ( r +
6
9
63. 1 2 5 / + z
3
— 25. To factor this polynomial, is the first step
2
RELATING
CONCEPTS
E X E
R C I S E S 65-70
FOR INDIVIDUAL OR GROUP WORK
The binomial x — y may be considered either as a difference of squares or a difference
of cubes. Work Exercises 65-70 in order.
6
6
65. Factor x — y by first factoring as a difference o f squares. Then factor further by
considering one of the factors as a sum o f cubes and the other factor as a difference
of cubes.
6
6
66. Based on your answer in Exercise 65, f i l l i n the blank with the correct factors so
that x — y is factored completely.
6
6
6
X
-
y6
= (
X
- y)(
X
+
67. Factor x — y by first factoring as a difference o f cubes. Then factor further by
considering one o f the factors as a difference o f squares.
6
6
68. Based on your answer in Exercise 67, f i l l i n the blank with the correct factor so that
x — y is factored.
6
6
69. Notice that the factor you wrote in the blank in Exercise 68 is a fourth-degree
polynomial, while the two factors you wrote i n the blank in Exercise 66 are both
second-degree polynomials. What must be true about the product o f the two factors
you wrote in the blank i n Exercise 66? Verify this.
70. I f you have a choice o f factoring as a difference o f squares or a difference of cubes,
how should you start to more easily obtain the completely factored form o f the
polynomial? Base the answer on your results i n Exercises 65-69.
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represents a copyright violation.
40
CHAPTER R
Review of Basic Concepts
•
FACTORING BY SUBSTITUTION
EXAMPLE 7
Factor each polynomial.
(a) 6 z
4
-
(c) (2a -
2
13z - 5
(b) 10(2a -
l)
2
19(2a -
1) -
15
3
l) + 8
Solution
2
2
2
(a) Replace z with u, so u
4
6z -
2
4
= (z )
2
13z - 5 = 6/r
= z.
- 1 3 H - 5
Remember to make the L = (2u final substitution.
P^*"^
,
1
5) (3M +
= (2z'
1)
Use F O I L to factor
5) (3?" + 1)
Replace i< with r .
(Some students prefer to factor this type of trinomial directly using trial and
error with F O I L . )
2
(b) I0(2a -
l) 2
= 10H -
~ 5 = ^ ^ —
Replace uwith 2 a - 1 .
(
5
U
+
19(2a -
19w -
3 ) ( 2 M
= [5(2 ,
1) -
Replace 2a •
'
with w
Tact or.
5 )
I ) + 3][2(2a
(
= (10a - 5 + 3) (4a -
(c)
15
15
1) -
Let it
5]
2 - 5)
2a - 1.
Disiributivc properly
= (10a
2) (4a - 7)
Add.
= 2(5a -
l)(4a - 7)
Factor out the common i act or.
3
(2a
Let 2a -
1
I) + 8 = n + 8
3
= u +2
s
Factor,
= (u + 2 ) ( « - 2« + 4)
2
= [(2a -
1 - u.
Write as a sum o! cubes.
1) + 2] [(2a
- I)
2
- 2(2a -
1) + 4]
L e t = 2a - L
2
= (2a + 1) (4a - 4a + 1 - 4a + 2 + 4)
Add: multiply.
=
(2a +
l)(4a
2
-
8a + 7)
Combine like terms.
NOW TRY EXERCISES 79, 81, AND 97. <
Exercises
Factor out the greatest common factor from each polynomial. See Examples 1 and 2
1. 12w + 60
4
4. 9<-. + 81:
3. S i + 24*
6. 5h-j + hj
7. - 4 p V - 2pV
y
A
9. 4k-m + U
3
2. 15r - 27
5. xy - 5xy-
8. - 3 z V -
3
12. 4(v - 2) + 3( v - 2)
14. (3z + 2) (z + 4) - (z + 6) (z + 4)
13. (5r - 6)(r + 3) - (2r - l)(r + 3)
:
35rV
2
11. 2(a + b) + 4m(u + b)
15. 2(m - 1) - 3(m - 1) + 2(m - l )
18zV
10. 2 8 r V + 7r s -
12ArV
3
3
16. 5(a + 3) - 2(a + 3) + (a + 3)
2
R.4 Factoring Polynomials
41
5
17. Concept Check When directed to completely factor the polynomial 4x~y — Kxy',
a student wrote 2xyH2x\< - 4). When the teacher did not give him full credit, he
complained because when his answer is multiplied out. the result is the original
polynomial. Give the correct answer.
2
Factor each polynomial by grouping. See Example 2.
18. lOafc - 6b + 35a - 21
2
2
19. 6st + 9t - 10s - 15
l
4
20. 15 - 5m - 3 r + m r
2
21. 2m* + 6 - am - 3a
2
2
22. 20c - 8x + 5pz - 2px
2
2
23. p q
2
- 10 - 2a + 5p
2
24. Concept Check Layla factored 16a - 40a - 6a + 15 by grouping and obtained
(8a - 3) (2a - 5). Jamal factored the same polynomial and gave an answer of
(3 - 8a) (5 - 2a). Which answer is correct?
Factor each trinomial, if possible. See Examples 3 and 4.
2
26. 8/i - 2h - 21
27. 3m + 14m + 8
2
29. 15/r + 24p + 8
2
30. 9x + 4.x - 2
25. 6a - 11a + 4
28. 9y - 18y + 8
31.
12a
3
+ 10a
2
3
- 42a
2
37. 12,r - xy - y
5
2
32. 36JC + 18x
2
34. 14m + llmr - 15r
2
2
38. 30a + am - m
2
2
33. 6k
2
2
4
2
- 4x
2
2
4
2
42. 16p - 40> + 25
2
2
2
2
45. 4x y + 28rv + 49
46. 9m n + \2mn + 4
47. (a - 3fc) - 6(a - 36) + 9
48. (2p + qf - 10(2p + a) + 25
2
49. Concept Check
Column II.
Match each polynomial in Column I with its factored form in
I
II
2
2
A. (x + 5y)(* - 5y)
2
B. (v + 5y)
2
C . (r - 5y)
2
(a) x + \0xy + 25y
2
(b) -V - lOtry + 25v
2
(c) x - 25y
2
2
44. 20p - lOOpa + 125a
2
2
3
39. 24a + 10a fc - 4a 6
41. 9m - 12m + 4
2
6p
36. 12s + list - 5r
2
43. 32a + 48a6 + 186
2
+ 5kp 2
35. 5a - lab - 6b
40. 18x + 15.T I - 15xh
2
2
2
2
2
2
(d) 25y - x
50. Concept Check
Column II.
D. (5y + x) (5y - x)
Match each polynomial in Column I with its factored form in
I
II
3
2
(a) 8x - 27
A. (3 - 2r) (9 + 6x + 4x )
3
2
(b) 8r + 27
(c) 27 - 8x
B. (2x - 3) (4.r + 6x + 9)
3
2
C . (2x + 3) (4x - 6x + 9)
Factor each polynomial. See Examples 5 and 6. (In Exercises 53 and 54, factor over the
rational numbers.)
2
2
51. 9a - 16
2
52. 16a - 25
53. 36* - —
25
4
2
54. 100v
55. 25s" - 9r
49
57. (a + b) - 16
2
4
60. m - 81
66. 27
3
Z
+ 729y
2
3
69. 21 - (m + 2nf
4
59. p - 625
3
64. 8m - 27n
3
4
62. r + 27
3
63. 125.r - 27
2
56. 36z - 81v
58. (p - 2a) - 100
61. 8 - a
3
2
3
3
67. (r + 6) - 216
70. 125 - (4a - bf
65. 27y" + 125c
3
6
68. (* + 3) - 27
SECTION 6.5
o
Brain Busters
49. 2(x 51. 5(3x -
Solving Equations by Factoring
351
Solve each equation.
2
l ) - l(x -
1) -
2
l ) + 3 = -16(3x 2
53. (2x - 3 ) =
2
50. 4 ( 2 * + 3 ) - (2x + 3) - 3 = 0
15 = 0
2
52. 2(x + 3 ) = 5{x + 3) - 2
1)
2
\6x
2
54. 9 x = (5x + 2 )
2
Solve each problem. See Examples 7 and 8.
x+4
2
55. A garden has an area o f 320 f t . Its length is
4 ft more than its width. What are the dimensions o f the garden? (Hint: 320 = 16 • 20)
H
56. A square mirror has sides measuring 2 ft
less than the sides o f a square painting. I f
the difference between their areas is 32 ft ,
find the lengths o f the sides o f the mirror
and the painting.
x
H
2
© 57. The base o f a parallelogram is 7 ft more than
the height. I f the area o f the parallelogram is
60 ft , what are the measures o f the base and
the height?
2
58. A sign has the shape o f a triangle. The
length o f the base is 3 m less than the height.
What are the measures o f the base and the
height i f the area is 44 m ?
2
2
59. A farmer has 300 ft o f fencing and wants to enclose a rectangular area o f 5000 f t . What
dimensions should she use? (Hint: 5000 = 50 • 100)
2
60. A rectangular landfill has an area o f 30,000 ft . Its length is 200 ft more than its width.
What are the dimensions o f the landfill? (Hint: 30,000 = 300 • 100)
61. Find two consecutive integers such that the sum o f their squares is 6 1 .
62. Find two consecutive integers such that their product is 72.
63. A box with no top is to be constructed from a piece o f cardboard whose length measures
6 in. more than its width. The box is to be formed by cutting squares that measure 2 in.
on each side from the four corners and then folding up the sides. I f the volume o f the box
w i l l be 110 i n . , what are the dimensions o f the piece o f cardboard?
3
w+6
J 2
2
2l_
2
2
12
2
2r
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represents a copyright violation.
Solving Quadratic Equations
C o m p l e t e the Square
Complete the Square to Solve the Equations
Worksheet # 4
1 .}Zx'-Ax+
3 = 0
2.)^+
Z^^-Ax-
32 = 0
4.)
15 = 0
6 . ) ^ - 6 x - 27 = 0
5.)^+
Qn+
Name:
10c-11 =
2
?_x - 2x- 48 = 0
7 . ) W - 12«+ 32 = 0
8 . ) £ « - 1 2 w + 27 =
9.)3/f- 16»+ 63 = 0
10.) 2^+ 18x+ 65 =
http://worksheetplace.com©
2
/
Solving Quadratic Equations
C o m p l e t e the Square
Complete the Square to Solve the Equations
Name:.
Worksheet # 6
1 .J^x -16x+ 39 = 0
2.)
3 . ) i ^ - 1 6 x + 39 = 0
4.) %yf- 14/i+
5.)y^-6x-
6.)%rf-
2
55 = 0
2
2
3 x - 14x+ 13 = 0
10«-
$x*+4x-?.\
33 = 0
11=0
7.)^JC +8X-9 = 0
8.)
9.)^e- 10x+21
10.)3« + 18«+ 56 =
=0
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2
=0