Download Factoring easy trinomials

Factoring Trinomials
of the Type x 2 ± bx ± c
9-5
9-5
1. Plan
GO for Help
What You’ll Learn
Check Skills You’ll Need
• To factor trinomials
List all of the factors of each number. 1–8. See back of book.
. . . And Why
1. 24
2. 12
3. 54
4. 15
5. 36
6. 56
7. 64
8. 96
To factor trinomials like
h 2 - 4hk - 77k 2, as in
Example 4
Skills Handbook p. 755
Objectives
1
Examples
1
2
3
4
1
Factoring Trinomials
Trinomials
1 1 Factoring
Part
(x + 3)(x + 5) = x 2 + 5x + 3x + 3 ? 5
(5 + 3)x
=
x2 +
8x
+
15
Notice that the coefficient of the middle term 8x is the sum of 3 and 5. Also the
constant term 15 is the product of 3 and 5. To factor a trinomial of the form
x 2 + bx + c, you must find two numbers that have a sum of b and a product of c.
The next example shows how to use a table to list the factors of the constant term c
and how to add the factors until the sum is the middle term b.
1
Factoring
x2
Factoring a trinomial that shows
no special pattern involves a
process of identifying possible
pairs of factors and then verifying
which pair will produce the
correct sum or difference.
More Math Background: p. 492D
Lesson Planning and
Resources
See p. 492E for a list of the
resources that support this lesson.
± bx ± c
Factor x 2 + 7x + 12.
PowerPoint
Find the factors of 12. Identify the pair that has a sum of 7.
For: Factoring Activity
Use: Interactive Textbook, 9-5
Factoring x2 + bx + c
Factoring x2 - bx + c
Factoring Trinomials With a
Negative c
Factoring Trinomials With
Two Variables
Math Background
In earlier courses, you learned how to find the factors of whole numbers like 15.
Since 3 3 5 = 15, 3 and 5 are factors of 15. You can also find the factors of some
trinomials. Consider the product below.
EXAMPLE
To factor trinomials
Bell Ringer Practice
Factors of 12
Sum of Factors
1 and 12
13
2 and 6
8
Skills Handbook: p. 754,
3 and 4
7✓
Example 2
Check Skills You’ll Need
For intervention, direct students to:
x2 + 7x + 12 = (x + 3)(x + 4)
Check x2 + 7x + 12 0 (x + 3)(x + 4)
= x 2 + 4x + 3x + 12
= x 2 + 7x + 12 ✓
Quick Check
1 Factor each expression. Check your answer.
a. g 2 + 7g + 10
b. v 2 + 21v + 20
(g ± 5)(g ± 2)
(v ± 20)(v ± 1)
c. a 2 + 13a + 30
(a ± 10)(a ± 3)
Lesson 9-5 Factoring Trinomials of the Type x2 1 bx 1 c
Special Needs
Below Level
L1
Have students check their answers using FOIL,
drawing area models, or using algebra tiles, so they
can see that factoring a trinomial is the inverse of
multiplying two binomials.
learning style: verbal
519
L2
Suggest that students always check their factoring by
multiplying the factors.
learning style: verbal
519
2. Teach
Some factorable trinomials have a negative middle term and a positive constant
term. If the middle term is negative, you need to inspect the negative factors of c to
find the factors of the trinomial.
Guided Instruction
3
EXAMPLE
2
4
Factoring x 2 – bx ± c
Factor d 2 - 17d + 42.
Math Tip
Since the middle term is negative, find the negative factors of 42. Identify the pair
that has a sum of -17.
Choosing the correct signs of the
factors is very important. To
emphasize this, have students
multiply (m + 3) and (m - 9).
EXAMPLE
EXAMPLE
Factors of 42
Error Prevention
It is a common error to forget to
write the last variable in the
factors. Encourage students to
write the parentheses, and place
both variables inside the
parentheses before factoring.
Remind students to leave spaces
for the numbers.
Sum of Factors
⫺1 and ⫺42
⫺43
⫺2 and ⫺21
⫺23
⫺3 and ⫺14
⫺17 ✓
d 2 - 17d + 42 = (d - 3)(d - 14)
Quick Check
2 Factor each expression.
a. k 2 - 10k + 25
(k – 5)(k – 5)
b. x 2 - 11x + 18
(x – 2)(x – 9)
c. q 2 - 15q + 36
(q – 12)(q – 3)
When you factor trinomials with a negative constant, you will need to inspect pairs
of positive and negative factors of c.
PowerPoint
Additional Examples
3
Factor each expression.
EXAMPLE
Factoring Trinomials With a Negative c
a. Factor m 2 + 6m - 27.
x2
1 Factor
+ 8x + 15.
(x ± 3)(x ± 5)
Identify the pair of factors of -27 that has a sum of 6.
Factors of 27
2 Factor c 2 - 9c + 20.
1 and ⫺27
(c – 5)(c – 4)
3 a. Factor x 2 + 13x - 48.
(x ± 16)(x – 3)
b. Factor n2 - 5n - 24.
(n ± 3)(n – 8)
Sum of Factors
⫺26
27 and ⫺1
26
3 and ⫺9
⫺6
9 and ⫺3
6✓
m 2 + 6m - 27 = (m - 3)(m + 9)
4 Factor d 2 + 17dg - 60g 2.
b. Factor p 2 - 3p - 18.
(d – 3g)(d ± 20g)
Identify the pair of factors of -18 that has a sum of -3.
Resources
• Daily Notetaking Guide 9-5 L3
• Daily Notetaking Guide 9-5—
L1
Adapted Instruction
Factors of 18
1 and ⫺18
17
⫺6 and 3
⫺3 ✓
p2 - 3p - 18 = (p + 3)(p - 6)
Quick Check
520
3 Factor each expression.
a. m 2 + 8m - 20
(m ± 10)(m – 2)
b. p2 - 3p - 40
(p – 8)(p ± 5)
c. y 2 - y - 56
(y ± 7)(y – 8)
Chapter 9 Polynomials and Factoring
Advanced Learners
English Language Learners ELL
L4
Challenge students to write a trinomial of the type
x2 + bx + c that cannot be written as (x + m)(x + n),
where m and n are integers.
520
⫺17
18 and ⫺1
Closure
Ask students to explain how to
determine what numbers are
used in the binomial factors when
factoring expressions of the type
x 2 + bx + c. You must find two
numbers that have a product of c
and a sum of b.
Sum of Factors
learning style: verbal
Have students point out the number in a trinomial for
which they need to find factors, and which number
must be the sum of the factors. Have them work
backwards from a factored answer to the original
trinomial.
learning style: visual
You can also factor some trinomials that have more than one variable. Consider
the product ( p + 10q)( p + 3q).
( p + 10q)( p + 3q) = p2 + 3pq + 10pq + 10q ? 3q
Assignment Guide
(3 + 10)pq
= p2 +
13pq + 30q 2
1 A B 1-58
You can see that the first term is the square of the first variable, the middle
term includes both variables, and the last term includes the square of the
second variable.
4
Factoring Trinomials With Two Variables
EXAMPLE
h2
Factor
- 4hk -
77k2.
Factors of 77
Find the factors of -77. Identify
the pair that has a sum of -4.
1 and ⫺77
4 Factor each expression.
a. x 2 + 11xy + 24y 2
(x ± 8y)(x ± 3y)
EXERCISES
Test Prep
Mixed Review
65-71
72-86
Error Prevention!
76
⫺4 ✓
Exercises 21–29 Suggest students
circle the signs in each expression
before factoring.
h2 - 4hk - 77k2 = (h + 7k)(h - 11k)
Quick Check
59-64
To check students’ understanding
of key skills and concepts, go over
Exercises 14, 22, 42, 54, 55.
⫺76
7 and ⫺11
C Challenge
Homework Quick Check
Sum of Factors
77 and ⫺1
3. Practice
b. v 2 + 2vw - 48w 2
(v ± 8w)(v – 6w)
c. m 2 - 17mn - 60n 2
(m – 20n)(m ± 3n)
For more exercises, see Extra Skill and Word Problem Practice.
Practiceand
andProblem
ProblemSolving
Solving
Practice
GPS Guided Problem Solving
Practice by Example
Examples 1, 2
(pages 519, 520)
GO for
Help
1.
t2
L4
Enrichment
Complete.
+ 7t + 10 = (t + 2)(t + j) 5
3. x 2 - 8x + 7 = (x - 1)(x - j) 7
2.
y2
- 13y + 36 = (y - 4)(y - j) 9
4. x 2 + 9x + 18 = (x + 3)(x + j) 6
L2
Reteaching
L1
Adapted Practice
Practice
Name
Class
Practice 9-5
Factor each expression. Check your answer. 5–13. See margin.
5.
(page 520)
+ 4r + 3
8.
y2
+ 6y + 8
11.
k2
- 16k + 28
6.
+ 21d + 38
(d ± 19)(d ± 2)
Complete.
14.
Example 3
r2
17.
19.
d2
m2
k2
n2
- 3n + 2
9.
x2
- 2x + 1
12.
w2
+ 6w + 5
7.
15. - 13t + 42
(t – 7)(t – 6)
- 8k - 9 = (k + 1)(k - j) 9
20.
10.
p2
13.
m2
Factor each expression.
+ 5k + 6
q2
24.
h2
22. q 2 - 2q - 8
+ 16h - 17
25.
27.
- 13m - 30
(m ± 2)(m – 15)
m2
x2
- 14x - 32
28.
+ 3p - 54
(p – 6)(p ± 9)
p2
Exercises
6. x2 + 12x + 32
9. a2 + 3a + 2
+ 19p + 18
15. x2 + 14x + 45
16. a2 + 7a + 12
17. x2 + 13x + 22
18. x2 + 3x - 4
19. x2 - 8x + 12
20. x2 + 7x - 18
21. n2 - 7n + 10
- 9m + 8
22. s2 - 5s - 14
23. x2 - 9x + 8
24. x2 - 2x - 24
25. x2 - 6x - 27
26. x2 - 16x - 36
27. x2 + 7x + 10
28. x2 - 3x - 28
29. m2 - 4m - 21
30. x2 - 2x - 15
31. x2 - 5x - 24
32. b2 - 4b - 60
33. x2 - 3x - 18
34. m2 + 7m + 10
35. n2 - n - 72
36. k2 - 6k + 5
37. x2 + 9x + 20
38. x2 - 10x + 9
39. x2 - 8x + 16
40. d2 - 4d + 3
41. b2 - 26b + 48
42. n2 - 15n + 26
43. n2 - n - 6
44. z2 - 14z + 49
45. x2 + 7x + 12
46. x2 - 18x + 17
47. x2 + 16x + 28
48. t2 - 6t - 27
49. b2 + 4b - 12
50. d2 + 11d + 18
51. x2 + x - 20
52. x2 - 13x + 42
53. x2 + x - 6
54. x2 + 4x - 21
55. a2 + 2a - 35
56. h2 + 7h - 18
57. x2 + 3x - 10
58. p2 - 12p - 28
59. y2 + 6y - 55
60. b2 + 3b - 4
61. x2 + 2x - 63
62. x2 - 2x - 8
63. x2 - 11x - 60
64. r2 + 2r - 35
65. c2 - 3c - 10
66. x2 + 8x + 15
67. x2 - 8x + 15
68. n2 - 23n + 60
69. c2 + 3c - 10
70. x2 - 9x + 14
71. x2 - 10x + 24
72. x2 + 6x - 27
73. y2 - 16y + 64
74. n2 + 10n + 25
75. r2 - 14r - 51
76. x2 + 3x - 40
77. x2 - x - 42
78. n2 - 2n - 63
79. a2 + 7a + 6
80. x2 - 14x + 48
81. x2 - 11x + 28
82. n2 + 16n - 36
83. n2 - 4n - 21
84. y2 + 16y - 17
23. y2 + y - 20
+ 6d - 40
29.
- 15p - 54
(p ± 3)(p – 18)
p2
Lesson 9-5 Factoring Trinomials of the Type x2 1 bx 1 c
pages 521–523
3. y2 + 6y + 8
14. b2 + 9b + 14
+ 3q - 18 = (q - 3)(q + j) 6
26.
8. x2 - 5x + 6
13. x2 + 5x - 14
- 2v - 24 = (v + 4)(v - j) 6
d2
5. s2 - 4s - 5
7. x2 - 9x + 20
12. n2 + n - 6
Factor each expression. 21–26. See margin.
21. x 2 + 3x - 4
2. d2 + 8d + 7
4. b2 - 2b - 3
11. d2 + 6d + 5
q2
v2
1. x2 + 8x + 16
10. p2 - 8p + 7
16.
- 18q + 45
(q – 15)(q – 3)
t2
+ 3m - 10 = (m - 2)(m + j) 5 18.
k2
L3
Date
Factoring Trinomials of the Type x2 + bx + c
© Pearson Education, Inc. All rights reserved.
A
L3
521
8. (y ± 4)(y ± 2)
12. (w ± 5)(w ± 1)
23. (y ± 5)(y – 4)
5. (r ± 3)(r ± 1)
9. (x – 1)(x – 1)
13. (m – 1)(m – 8)
24. (h ± 17)(h – 1)
6. (n – 2)(n – 1)
10. (p ± 18)(p ± 1)
21. (x ± 4)(x – 1)
25. (x – 16)(x ± 2)
7. (k ± 3)(k ± 2)
11. (k – 14)(k – 2)
22. (q – 4)(q ± 2)
26. (d ± 10)(d – 4)
521
4. Assess & Reteach
Example 4
(page 521)
33. (t ± 9v)(t – 2v)
PowerPoint
Lesson Quiz
Factor each expression.
1.
c2
+ 6c + 9 (c ± 3)(c ± 3)
2.
x2
- 11x + 18 (x – 2)(x – 9)
3.
g2 -
34. (x ± 7y)(x ± 5y)
- bx - c
(+)(-)
A. (m + n)(3m + n)
B. (m + 3n)(m + n)
x2
+ 8xy +
15y2
A. A x +
15y2 B (x
+ 1)
B. (x + 5y)(x + 3y)
Factor each expression. 33–38. See left.
33. t 2 + 7tv - 18v2
34. x2 + 12xy + 35y2
35. p2 - 10pq + 16q2
36. m2 - 3mn - 54n2
37. h2 + 18hj + 17j2
38. x2 - 10xy - 39y2
38. ( x – 13y)(x ± 3y)
B
Apply Your Skills
B
Open-Ended Find three different values to complete each expression so that it can
be factored into the product of two binomials. Show each factorization.
39. x 2 - 3x - j
40. x 2 + x - j
41. x2 + jx + 12
39–42. See margin.
42. Writing Suppose you can factor x 2 + bx + c into the product of two binomials.
a. Explain what you know about the factors if c . 0.
b. Explain what you know about the factors if c , 0.
Have students make posters
displaying how the signs in a
trinomial affect the binomial
factors. Instruct them to write
examples using the standard form
of a quadratic equation with
a = 1. Encourage them to use
one color for plus signs and
another color for minus signs.
Here is an example:
+ bx - c
(+)(-)
31. m2 + 4mn + 3n2 B
37. (h ± 17j)(h ± j)
Alternative Assessment
x2
B. ( p + 9) A p + q 2 B
36. ( m – 9n)(m ± 6n)
5. m2 - 2mn + n2
(m – n)(m – n)
x2
A. ( p + 9q)( p + q)
32.
2g - 24 ( g – 6)( g ± 4)
x 2 - bx + c
(-)(-)
30. p2 + 10pq + 9q 2 A
35. ( p – 8q)(p – 2q)
4. y 2 + y - 110 (y ± 11)(y – 10)
x 2 + bx + c
(+)(+)
Choose the correct factoring for each expression.
Factor each expression. 43–51. See margin.
43. k 2 + 10k + 16
44. m2 + 10m - 24
45. n2 + 10n - 56
46. g2 + 20g + 96
47. x2 + 8x - 65
48. t2 + 28t + 75
49. x2 - 11x - 42
50. k2 + 23k + 42
51. m2 + 14m - 51
52. x 2 + 29xy + 100y2
53. t 2 - 10t - 75
54. d 2 - 19de + 48e2
(t – 15)(t ± 5)
(x ± 25y)(x ± 4y)
(d – 16e)(d – 3e)
Write the standard form for each of the polynomials modeled below. Then factor
each expression.
55. 4x2 ± 12x ± 5;
(2x ± 1)(2x ± 5)
GPS 55.
56.
56. 6x2 ± 13x ± 6;
(3x ± 2)(2x ± 3)
4x2
2x
GO
nline
Homework Help
Visit: PHSchool.com
Web Code: ate-0905
6x2
4x
9x
6
10x
5
57. Critical Thinking Let x 2 - 12x - 28 = (x + a)(x + b).
a–b. See margin p. 523.
a. What do you know about the signs of a and b?
b. Suppose Δa« . Δb«. Which number, a or b, is a negative integer? Explain.
58. Critical Thinking Let x 2 + 12x - 28 = (x + a)(x + b).
a–b. See margin p. 523.
a. What do you know about the signs of a and b?
b. Suppose Δa« . Δb«. Which number, a or b, is a negative integer? Explain.
C
pages 521–523
Challenge
Sample n6 + n3 - 56 = n 3 + 3 + n3 - 56
= An3 + 8B An3 - 7B
Exercises
39–41. Answers may vary.
Samples are given.
39. 18; (x – 6)(x ± 3)
28; (x – 7)(x ± 4)
10; (x – 5)(x ± 2)
40. 12; (x ± 4)(x – 3)
2; (x ± 2)(x – 1)
20; (x ± 5)(x – 4)
41. 7; (x ± 4)(x ± 3)
8; (x ± 6)(x ± 2)
13; (x ± 12)(x ± 1)
522
Factor each trinomial. 59–61. See margin p. 523.
522
59. x12 + 12x6 + 35
60. t 8 + 5t 4 - 24
61. r 6 - 21r 3 + 80
62. m10 + 18m5 + 17
(m5 ± 17)(m5 ± 1)
63. x12 - 19x6 - 120
(x6 – 24)(x6 ± 5)
64. p6 + 14p3 - 72
(p3 – 4)(p3 ± 18)
Chapter 9 Polynomials and Factoring
42a. Factors contain the
same operation.
b. Factors contain
opposite operations.
43. (k ± 2)(k ± 8)
44. (m – 2)(m ± 12)
48. (t ± 3)(t ± 25)
45. (n – 4)(n ± 14)
49. (x – 14)(x ± 3)
46. (g ± 12)(g ± 8)
50. (k ± 21)(k ± 2)
47. (x – 5)(x ± 13)
51. (m – 3)(m ± 17)
Test Prep
StandardizedTest
TestPrep
Prep
Resources
Multiple Choice
65. Which of the following is NOT a factor of 72? B
A. 12
B. 16
C. 18
66. Which value of b would make the expression
F. 5
G. 4
H. 3
x2
D. 24
+ bx - 36 factorable? F
J. 2
67. Which value of c would NOT make x2 + 10x + c factorable? D
A. 25
B. 24
C. 21
D. 18
68. Which of the following products is J
represented by the area model?
F. (x + 1)(x + 18)
G. (x + 3)(x + 6)
H. (x - 1)(x - 18)
J. (x + 2)(x + 9)
69. Which of the following shows the
factors of g 2 + 18g + 72? A
A. (g + 6)(g + 12)
C. (g - 6)(g - 12)
x2
9x
Exercise 69 Tell students they can
quickly eliminate some of the
incorrect answer choices. Point
out that since all signs are
positive in the trinomial, all signs
are positive in the factors.
2x
18
59. (x6 ± 7)(x6 ± 5)
60. (t 4 ± 8)(t 4 – 3)
B. (g + 18)(g + 72)
D. (g - 18)(g - 72)
61. (r 3 – 16)(r 3 – 5)
70. Which of the following shows the factors of n 2 - 15g + 50? F
F. (n - 5)(n - 10)
G. (n + 5)(n - 10)
H. (n - 15)(n + 50)
J. (n + 15)(n - 50)
Short Response
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 545
• Test-Taking Strategies, p. 540
• Test-Taking Strategies with
Transparencies
71. Explain how to factor the trinomial x 2 - 18x - 40 and state the factors.
See margin.
71. [2] Find a pair of factors
of –40 that has a sum
of –18: –20 and 2.
x2 – 18x – 40 ≠
(x – 20)(x ± 2)
[1] correct explanation
with incorrect
factoring OR
incorrect explanation
with correct factoring
Mixed
Mixed Review
Review
GO for
Help
Lesson 9-4
Lesson 7-4
Simplify each product. 72–77. See margin.
72. x2 ± 8x ± 16
72. (x + 4)(x + 4)
73. (w - 6)(w - 6)
74. (r - 5)(r + 5)
73. w2 – 12w ± 36
75. (2q + 7)(2q + 7)
76. (8v - 2)(8v + 2)
77. (3a - 9)(3a - 9)
74. r 2 – 25
78. (3a - 5)(3a + 5)
79. (6t + 9)(6t + 9)
80. (2x + 8y)(2x - 8y)
2
9a2 – 25
4x2 – 64y2
36t ± 108t ± 81
81. You start with $40 in your bank account and deposit $18 each week. At the same
time, your friend starts with $220 but withdraws $12 each week. When will your
accounts have the same balance? 6 weeks
82. The sum of the two numbers is 42. The smaller number is 63 less than twice the
larger number. Find both numbers. 7, 35
83. Sales A department store sells two types of DVD players. Total sales of players
for the year were $16,918.71. The total number of players sold was 129. The basic
model costs $119.99. The deluxe model costs $149.99.
a. Find the number sold of each type of player. 81 basic players, 48 deluxe
b. What were the sales for the basic player? $9719.19
players
Lesson 7-1
75. 4q2 ± 28q ± 49
76. 64v2 – 4
77. 9a2 – 54a ± 81
84.
(1, 1) 1
3
y = 2x + 3
lesson quiz, PHSchool.com, Web Code: ata-0905
85. y = x + 4
y = 0.5x + 5
85.
86. 2x + 4y = 12
4
x -y = 3
2
b. Since the middle term is
negative, the number
with the larger absolute
value must be negative.
Therefore, a must be a
negative integer.
58a. The signs of a and b
must be opposite.
b. Since the middle term is
positive, the number
with the larger absolute
value must be positive.
Therefore, b is a
negative integer.
(2, 6)
O
523
86.
57a. The signs of a and b
must be opposite.
2
y
6
Lesson 9-5 Factoring Trinomials of the Type x2 1 bx 1 c
x
O
2
Solve each system by graphing. 84–86. See margin.
84. y = -2x - 1
y
2
4
x
y
(4, 1)
2
x
O
2
4
6
2
4
523