Download Use the Distributive Property to factor each polynomial. 35. 12x +

Study Guide and Review - Chapter 8
Use the Distributive Property to factor each polynomial.
35. 12x + 24y
SOLUTION: Factor
The greatest common factor of each term is 2· 3 or 12.
12x + 24y = 12(x + 2y)
2
2
36. 14x y − 21xy + 35xy
SOLUTION: Factor
The greatest common factor of each term is 7xy.
2
2
14x y − 21xy + 35xy = 7xy(2x − 3 + 5y)
3
37. 8xy − 16x y + 10y
SOLUTION: Factor.
The greatest common factor of each term is 2y.
3
3
8xy − 16x y + 10y = 2y(4x − 8x + 5)
2
38. a − 4ac + ab − 4bc
SOLUTION: Factor by grouping.
2
39. 2x − 3xz − 2xy + 3yz
SOLUTION: 40. 24am − 9an + 40bm − 15bn
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SOLUTION: Page 1
Study Guide and Review - Chapter 8
40. 24am − 9an + 40bm − 15bn
SOLUTION: Solve each equation. Check your solutions.
2
41. 6x = 12x
SOLUTION: Factor the trinomial using the Zero Product Property.
or The roots are 0 and 2. Check by substituting 0 and 2 in for x in the original equation.
and
The solutions are 0 and 2.
2
42. x = 3x
SOLUTION: Factor the trinomial using the Zero Product Property.
The roots are 0 and 3. Check by substituting 0 and 3 in for x in the original equation.
and
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The solutions are 0 and 3.
2
Page 2
Study
Guide
and are
Review
The
solutions
0 and-2.Chapter 8
2
42. x = 3x
SOLUTION: Factor the trinomial using the Zero Product Property.
The roots are 0 and 3. Check by substituting 0 and 3 in for x in the original equation.
and
The solutions are 0 and 3.
2
43. 3x = 5x
SOLUTION: Factor the trinomial using the Zero Product Property.
The roots are 0 and
. Check by substituting 0 and
in for x in the original equation.
and
The solutions are 0 and
.
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44. x(3x − 6) = 0
SOLUTION: Page 3
Study Guide and Review - Chapter 8
The solutions are 0 and 3.
2
43. 3x = 5x
SOLUTION: Factor the trinomial using the Zero Product Property.
The roots are 0 and
. Check by substituting 0 and
in for x in the original equation.
and
The solutions are 0 and
.
44. x(3x − 6) = 0
SOLUTION: Factor the trinomial using the Zero Product Property.
x(3x − 6) = 0
x = 0 or The roots are 0 and 2. Check by substituting 0 and 2 in for x in the original equation.
and
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The solutions are 0 and 2.
Page 4
Study
Guide
and are
Review
8
The
solutions
0 and- Chapter
.
44. x(3x − 6) = 0
SOLUTION: Factor the trinomial using the Zero Product Property.
x(3x − 6) = 0
x = 0 or The roots are 0 and 2. Check by substituting 0 and 2 in for x in the original equation.
and
The solutions are 0 and 2.
3
2
45. GEOMETRY The area of the rectangle shown is x − 2x + 5x square units. What is the length?
SOLUTION: 3
2
2
The area of the rectangle is x − 2x + 5x or x(x – 2x + 5). Area is found by multiplying the length by the width.
2
Because the width is x, the length must be x − 2x + 5.
Factor each trinomial. Confirm your answers using a graphing calculator.
2
46. x − 8x + 15
SOLUTION: In this trinomial, b = –8 and c = 15, so m + p is negative and mp is positive. Therefore, m and p must both be
negative. List the negative factors of 15, and look for the pair of factors with a sum of –8.
Factors of 15
–1, –15
–3, –5
Sum of –8
–16
–8
The correct factors are –3 and –5.
Check using a Graphing calculator.
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SOLUTION: 3
2
2
The
area of
theReview
rectangle
is x − 2x8 + 5x or x(x – 2x + 5). Area is found by multiplying the length by the width.
Study
Guide
and
- Chapter
2
Because the width is x, the length must be x − 2x + 5.
Factor each trinomial. Confirm your answers using a graphing calculator.
2
46. x − 8x + 15
SOLUTION: In this trinomial, b = –8 and c = 15, so m + p is negative and mp is positive. Therefore, m and p must both be
negative. List the negative factors of 15, and look for the pair of factors with a sum of –8.
Factors of 15
–1, –15
–3, –5
Sum of –8
–16
–8
The correct factors are –3 and –5.
Check using a Graphing calculator.
[– 10, 10] scl: 1 by [– 10, 10] scl: 1
2
47. x + 9x + 20
SOLUTION: In this trinomial, b = 9 and c = 20, so m + p is positive and mp is positive. Therefore, m and p must both be positive.
List the positive factors of 20, and look for the pair of factors with a sum of 9.
Factors of 20
1, 20
2, 10
4, 5
Sum of 9
21
12
9
The correct factors are 4 and 5.
Check using a Graphing calculator.
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[– 10, 10] scl: 1 by [– 10, 10] scl: 1
2
Page 6
Study
Guide
- Chapter
[– 10,
10]and
scl: Review
1 by [– 10,
10] scl: 18
2
47. x + 9x + 20
SOLUTION: In this trinomial, b = 9 and c = 20, so m + p is positive and mp is positive. Therefore, m and p must both be positive.
List the positive factors of 20, and look for the pair of factors with a sum of 9.
Factors of 20
1, 20
2, 10
4, 5
Sum of 9
21
12
9
The correct factors are 4 and 5.
Check using a Graphing calculator.
[– 10, 10] scl: 1 by [– 10, 10] scl: 1
2
48. x − 5x − 6
SOLUTION: In this trinomial, b = –5 and c = –6, so m + p is negative and mp is negative. Therefore, m and p must have different
signs. List the factors of –6, and look for the pair of factors with a sum of –5.
Factors of –6
–1, 6
1, –6
2, –3
–2, 3
Sum of –5
5
–5
–1
1
The correct factors are 1 and −6.
Check using a Graphing calculator.
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[– 10, 10] scl: 1 by [– 12, 8] scl: 1
Page 7
Study Guide and Review - Chapter 8
[– 10, 10] scl: 1 by [– 10, 10] scl: 1
2
48. x − 5x − 6
SOLUTION: In this trinomial, b = –5 and c = –6, so m + p is negative and mp is negative. Therefore, m and p must have different
signs. List the factors of –6, and look for the pair of factors with a sum of –5.
Factors of –6
–1, 6
1, –6
2, –3
–2, 3
Sum of –5
5
–5
–1
1
The correct factors are 1 and −6.
Check using a Graphing calculator.
[– 10, 10] scl: 1 by [– 12, 8] scl: 1
2
49. x + 3x − 18
SOLUTION: In this trinomial, b = 3 and c = –18, so m + p is positive and mp is negative. Therefore, m and p must have different
signs. List the factors of –18, and look for the pair of factors with a sum of 3.
Factors of –18
–1, 18
1, –18
–2, 9
2, –9
–3, 6
3, –6
Sum of 3
17
–17
7
–7
3
–3
The correct factors are –3 and 6.
Check using a Graphing calculator.
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Page 8
Study Guide and Review - Chapter 8
[– 10, 10] scl: 1 by [– 12, 8] scl: 1
2
49. x + 3x − 18
SOLUTION: In this trinomial, b = 3 and c = –18, so m + p is positive and mp is negative. Therefore, m and p must have different
signs. List the factors of –18, and look for the pair of factors with a sum of 3.
Factors of –18
–1, 18
1, –18
–2, 9
2, –9
–3, 6
3, –6
Sum of 3
17
–17
7
–7
3
–3
The correct factors are –3 and 6.
Check using a Graphing calculator.
[– 10, 10] scl: 1 by [– 14, 6] scl: 1
Solve each equation. Check your solutions.
2
50. x + 5x − 50 = 0
SOLUTION: The roots are –10 and 5. Check by substituting –10 and 5 in for x in the original equation.
and
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The solutions are –10 and 5.
Page 9
Study Guide and Review - Chapter 8
[– 10, 10] scl: 1 by [– 14, 6] scl: 1
Solve each equation. Check your solutions.
2
50. x + 5x − 50 = 0
SOLUTION: The roots are –10 and 5. Check by substituting –10 and 5 in for x in the original equation.
and
The solutions are –10 and 5.
2
51. x − 6x + 8 = 0
SOLUTION: The roots are 2 and 4. Check by substituting 2 and 4 in for x in the original equation.
and
The solutions are 2 and 4.
2
52. x + 12x + 32 = 0
SOLUTION: eSolutions
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Study Guide and Review - Chapter 8
The solutions are 2 and 4.
2
52. x + 12x + 32 = 0
SOLUTION: The roots are –8 and –4. Check by substituting –8 and –4 in for x in the original equation.
and
The solutions are –8 and –4.
2
53. x − 2x − 48 = 0
SOLUTION: The roots are –6 and 8. Check by substituting –6 and 8 in for x in the original equation.
and
The solutions are –6 and 8.
2
54. x + 11x + 10 = 0
SOLUTION: eSolutions Manual - Powered by Cognero
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Study Guide and Review - Chapter 8
The solutions are –6 and 8.
2
54. x + 11x + 10 = 0
SOLUTION: The roots are –10 and –1. Check by substituting –10 and –1 in for x in the original equation.
and
The solutions are –10 and –1.
55. ART An artist is working on a painting that is 3 inches longer than it is wide. The area of the painting is 154 square
inches. What is the length of the painting?
SOLUTION: Let x = the width of the painting. Then, x + 3 = the length of the painting.
Because a painting cannot have a negative dimension, the width is 11 inches and the length is 11 + 3, or 14 inches.
Factor each trinomial, if possible. If the trinomial cannot be factored, write prime .
2
56. 12x + 22x − 14
SOLUTION: In this trinomial, a = 12, b = 22 and c = –14, so m + p is positive and mp is negative. Therefore, m and p must have
different signs. List the factors of 12(–14) or –168 and identify the factors with a sum of 22.
Factors of –168
Sum 82
–2, 84
2, –84
–82
53
–3, 56
3, –56
–53
–4,- 42
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4, –42
–38
22
–6, 28
Study Guide and Review - Chapter 8
Because a painting cannot have a negative dimension, the width is 11 inches and the length is 11 + 3, or 14 inches.
Factor each trinomial, if possible. If the trinomial cannot be factored, write prime .
2
56. 12x + 22x − 14
SOLUTION: In this trinomial, a = 12, b = 22 and c = –14, so m + p is positive and mp is negative. Therefore, m and p must have
different signs. List the factors of 12(–14) or –168 and identify the factors with a sum of 22.
Factors of –168
Sum 82
–2, 84
2, –84
–82
53
–3, 56
3, –56
–53
38
–4, 42
4, –42
–38
22
–6, 28
The correct factors are –6 and 28.
2
So, 12x + 22x − 14 = 2(2x − 1)(3x + 7).
2
57. 2y − 9y + 3
SOLUTION: In this trinomial, a = 2, b = –9 and c = 3, so m + p is negative and mp is positive. Therefore, m and p must both be
negative.
2(3) = 6
There are no factors of 6 with a sum of –9. So, this trinomial is prime.
2
58. 3x − 6x − 45
SOLUTION: In this trinomial, a = 3, b = –6 and c = –45, so m + p is negative and mp is negative. Therefore, m and p must have
different signs. List the factors of 3(–45) or –135 and identify the factors with a sum of –6.
Factors of –135
Sum
134
–1, 135
1, –135
–134
42
–3, 45
3, –45
–42
22
–5, 27
5, –27
–22
6
–9, 15
9, –15
–6
The correct factors are –15 and 9.
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negative.
2(3) = 6
Guide and Review - Chapter 8
Study
There are no factors of 6 with a sum of –9. So, this trinomial is prime.
2
58. 3x − 6x − 45
SOLUTION: In this trinomial, a = 3, b = –6 and c = –45, so m + p is negative and mp is negative. Therefore, m and p must have
different signs. List the factors of 3(–45) or –135 and identify the factors with a sum of –6.
Factors of –135
Sum
134
–1, 135
1, –135
–134
42
–3, 45
3, –45
–42
22
–5, 27
5, –27
–22
6
–9, 15
9, –15
–6
The correct factors are –15 and 9.
2
So, 3x − 6x − 45 = 3(x − 5)(x + 3).
2
59. 2a + 13a − 24
SOLUTION: In this trinomial, a = 2, b = 13 and c = –24, so m + p is positive and mp is negative. Therefore, m and p must have
different signs. List the factors of 2(–24) or –48 and identify the factors with a sum of 13.
Factors of –48
Sum 47
–1, 48
1, –48
–47
22
–2, 24
2, –24
–22
13
–3, 16
3, –16
–13
8
–4, 12
4, –12
–8
2
–6, 8
6, –8
–2
The correct factors are –3 and 16.
2
So, 2a + 13a − 24 = (2a − 3)(a + 8).
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Solve each equation. Confirm your answers using a graphing calculator.
2
60. 40x + 2x = 24
Page 14
Study Guide and Review - Chapter 8
2
So, 3x − 6x − 45 = 3(x − 5)(x + 3).
2
59. 2a + 13a − 24
SOLUTION: In this trinomial, a = 2, b = 13 and c = –24, so m + p is positive and mp is negative. Therefore, m and p must have
different signs. List the factors of 2(–24) or –48 and identify the factors with a sum of 13.
Factors of –48
Sum 47
–1, 48
1, –48
–47
22
–2, 24
2, –24
–22
13
–3, 16
3, –16
–13
8
–4, 12
4, –12
–8
2
–6, 8
6, –8
–2
The correct factors are –3 and 16.
2
So, 2a + 13a − 24 = (2a − 3)(a + 8).
Solve each equation. Confirm your answers using a graphing calculator.
2
60. 40x + 2x = 24
SOLUTION: The roots are
and or –0.80 and 0.75 . 2
Confirm the roots using a graphing calculator. Let Y1 = 40x + 2x and Y2 = –24. Use the intersect option from the
CALC menu to find the points of intersection. eSolutions Manual - Powered by Cognero
Page 15
Study Guide and Review - Chapter 8
2
So, 2a + 13a − 24 = (2a − 3)(a + 8).
Solve each equation. Confirm your answers using a graphing calculator.
2
60. 40x + 2x = 24
SOLUTION: The roots are
and or –0.80 and 0.75 . 2
Confirm the roots using a graphing calculator. Let Y1 = 40x + 2x and Y2 = –24. Use the intersect option from the
CALC menu to find the points of intersection. [–5, 5] scl: 1 by [–5, 25] scl: 3
The solutions are
and [–5, 5] scl: 1 by [–5, 25] scl: 3
.
2
61. 2x − 3x − 20 = 0
SOLUTION: eSolutions
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by −2.5 and 4. Cognero
The Manual
roots are
Page 16
2
Confirm the roots using a graphing calculator. Let Y1 = 2x − 3x − 20 and Y2 = 0. Use the intersect option from
[–5, 5] scl: 1 by [–5, 25] scl: 3
[–5, 5] scl: 1 by [–5, 25] scl: 3
The
solutions
.
Study
Guide
and are
Review and - Chapter
8
2
61. 2x − 3x − 20 = 0
SOLUTION: The roots are
or −2.5 and 4. 2
Confirm the roots using a graphing calculator. Let Y1 = 2x − 3x − 20 and Y2 = 0. Use the intersect option from
the CALC menu to find the points of intersection. [–10, 10] scl: 1 by [–15, 5] scl: 1
The solutions are
[–10, 10] scl: 1 by [–15, 5] scl: 1
and 4.
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