Content check — Elementary Algebra, Chapter 7: Factoring

Be Prepared 7.1 §7.1 source ↗

Factor 56 into primes.
If you missed this problem, review Example 1.7.

Answer

2 \cdot 2 \cdot 2 \cdot 7

Be Prepared 7.2 §7.1 source ↗

Find the least common multiple of 18 and 24.
If you missed this problem, review Example 1.10.

Example 1.10 in the book ↗

Answer

72

Be Prepared 7.3 §7.1 source ↗

Simplify

−3 (6 a + 11)

.
If you missed this problem, review Example 1.135.

Example 1.135 in the book ↗

Answer

- 18 a - 33

Try It 7.1 §7.1 source ↗

Find the GCF of 48 and 80.

Answer

16

Try It 7.2 §7.1 source ↗

Find the GCF of 18 and 40.

Answer

2

Try It 7.3 §7.1 source ↗

Find the GCF:

12 x^{2}, 18 x^{3}

.

Answer

6 x^{2}

Try It 7.4 §7.1 source ↗

Find the GCF:

16 y^{2}, 24 y^{3}

.

Answer

8 y^{2}

Try It 7.5 §7.1 source ↗

Find the GCF:

6 a b^{4}, 8 a^{2} b

.

Answer

2 a b

Try It 7.6 §7.1 source ↗

Find the GCF:

9 m^{5} n^{2}, 12 m^{3} n

.

Answer

3 m^{3} n

Try It 7.7 §7.1 source ↗

Find the greatest common factor:

25 m^{4},

35 m^{3},

20 m^{2}

.

Answer

5 m^{2}

Try It 7.8 §7.1 source ↗

Find the greatest common factor:

14 x^{3},

70 x^{2},

105 x

.

Answer

7 x

Try It 7.9 §7.1 source ↗

Factor:

6 a + 24

.

Answer

6 (a + 4)

Try It 7.10 §7.1 source ↗

Factor:

2 b + 14

.

Answer

2 (b + 7)

Try It 7.11 §7.1 source ↗

Factor:

14 x + 14

.

Answer

14 (x + 1)

Try It 7.12 §7.1 source ↗

Factor:

12 p + 12

.

Answer

12 (p + 1)

Try It 7.13 §7.1 source ↗

Factor:

18 u - 36

.

Answer

18 (u - 2)

Try It 7.14 §7.1 source ↗

Factor:

30 y - 60

.

Answer

30 (y - 2)

Try It 7.15 §7.1 source ↗

Factor:

5 x^{2} - 25 x + 15

.

Answer

5 (x^{2} - 5 x + 3)

Try It 7.16 §7.1 source ↗

Factor:

3 y^{2} - 12 y + 27

.

Answer

3 (y^{2} - 4 y + 9)

Try It 7.17 §7.1 source ↗

Factor:

2 x^{3} + 12 x^{2}

.

Answer

2 x^{2} (x + 6)

Try It 7.18 §7.1 source ↗

Factor:

6 y^{3} - 15 y^{2}

.

Answer

3 y^{2} (2 y - 5)

Try It 7.19 §7.1 source ↗

Factor:

20 x^{3} -

10 x^{2} +

14 x

.

Answer

2 x (10 x^{2} - 5 x + 7)

Try It 7.20 §7.1 source ↗

Factor:

24 y^{3} -

12 y^{2} -

20 y

.

Answer

4 y (6 y^{2} - 3 y - 5)

Try It 7.21 §7.1 source ↗

Factor:

9 x y^{2} +

6 x^{2} y^{2} +

21 y^{3}

.

Answer

3 y^{2} (3 x + 2 x^{2} + 7 y)

Try It 7.22 §7.1 source ↗

Factor:

3 p^{3} -

6 p^{2} q +

9 p q^{3}

.

Answer

3 p (p^{2} - 2 p q + 3 q^{3})

Try It 7.23 §7.1 source ↗

Factor:

−16 z - 64

.

Answer

−16 (z + 4)

Try It 7.24 §7.1 source ↗

Factor:

−9 y - 27

.

Answer

−9 (y + 3)

Try It 7.25 §7.1 source ↗

Factor:

−4 b^{2} + 16 b

.

Answer

−4 b (b - 4)

Try It 7.26 §7.1 source ↗

Factor:

−7 a^{2} + 21 a

.

Answer

−7 a (a - 3)

Try It 7.27 §7.1 source ↗

Factor:

4 m (m + 3) -

7 (m + 3)

.

Answer

(m + 3) (4 m - 7)

Try It 7.28 §7.1 source ↗

Factor:

8 n (n - 4) +

5 (n - 4)

.

Answer

(n - 4) (8 n + 5)

Try It 7.29 §7.1 source ↗

Factor:

x y + 8 y + 3 x + 24

.

Answer

(x + 8) (y + 3)

Try It 7.30 §7.1 source ↗

Factor:

a b + 7 b + 8 a + 56

.

Answer

(a + 7) (b + 8)

Try It 7.31 §7.1 source ↗

Factor:

x^{2} + 2 x - 5 x - 10

.

Answer

(x - 5) (x + 2)

Try It 7.32 §7.1 source ↗

Factor:

y^{2} + 4 y - 7 y - 28

.

Answer

(y + 4) (y - 7)

7.1-1 §7.1 source ↗

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

8, 18

Answer

2

7.1-3 §7.1 source ↗

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

72, 162

Answer

18

7.1-5 §7.1 source ↗

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

10a, 50

Answer

10

7.1-7 §7.1 source ↗

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$3 x, 10 x^{2}$

Answer

x

7.1-9 §7.1 source ↗

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$8 w^{2}, 24 w^{3}$

Answer

8 w^{2}

7.1-11 §7.1 source ↗

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$10 p^{3} q, 12 p q^{2}$

Answer

2 p q

7.1-13 §7.1 source ↗

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$12 m^{2} n^{3},$ $30 m^{5} n^{3}$

Answer

6 m^{2} n^{3}

7.1-15 §7.1 source ↗

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$10 a^{3},$ $12 a^{2},$ $14 a$

Answer

2 a

7.1-17 §7.1 source ↗

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$35 x^{3},$ $10 x^{4},$ $5 x^{5}$

Answer

5 x^{3}

7.1-19 §7.1 source ↗

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$4 x + 20$

Answer

4 (x + 5)

7.1-21 §7.1 source ↗

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$6 m + 9$

Answer

3 (2 m + 3)

7.1-23 §7.1 source ↗

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$9 q + 9$

Answer

9 (q + 1)

7.1-25 §7.1 source ↗

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$8 m - 8$

Answer

8 (m - 1)

7.1-27 §7.1 source ↗

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$9 n - 63$

Answer

9 (n - 7)

7.1-29 §7.1 source ↗

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$3 x^{2} + 6 x - 9$

Answer

3 (x^{2} + 2 x - 3)

7.1-31 §7.1 source ↗

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$8 p^{2} + 4 p + 2$

Answer

2 (4 p^{2} + 2 p + 1)

7.1-33 §7.1 source ↗

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$8 y^{3} + 16 y^{2}$

Answer

8 y^{2} (y + 2)

7.1-35 §7.1 source ↗

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$5 x^{3} -$ $15 x^{2} +$ $20 x$

Answer

5 x (x^{2} - 3 x + 4)

7.1-37 §7.1 source ↗

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$12 x y^{2} +$ $18 x^{2} y^{2} -$ $30 y^{3}$

Answer

6 y^{2} (2 x + 3 x^{2} - 5 y)

7.1-39 §7.1 source ↗

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$−2 x - 4$

Answer

−2 (x + 2)

7.1-41 §7.1 source ↗

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$5 x (x + 1) +$ $3 (x + 1)$

Answer

(x + 1) (5 x + 3)

7.1-43 §7.1 source ↗

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$3 b (b - 2) -$ $13 (b - 2)$

Answer

(b - 2) (3 b - 13)

7.1-45 §7.1 source ↗

Factor by Grouping

In the following exercises, factor by grouping.

$x y + 2 y + 3 x + 6$

Answer

(y + 3) (x + 2)

7.1-47 §7.1 source ↗

Factor by Grouping

In the following exercises, factor by grouping.

$u v - 9 u + 2 v - 18$

Answer

(u + 2) (v - 9)

7.1-49 §7.1 source ↗

Factor by Grouping

In the following exercises, factor by grouping.

$b^{2} + 5 b - 4 b - 20$

Answer

(b - 4) (b + 5)

7.1-51 §7.1 source ↗

Factor by Grouping

In the following exercises, factor by grouping.

$p^{2} + 4 p - 9 p - 36$

Answer

(p - 9) (p + 4)

7.1-53 §7.1 source ↗

Mixed Practice

In the following exercises, factor.

$−20 x - 10$

Answer

−10 (2 x + 1)

7.1-55 §7.1 source ↗

Mixed Practice

In the following exercises, factor.

$3 x^{3} -$ $7 x^{2} +$ $6 x -$ $14$

Answer

(x^{2} + 2) (3 x - 7)

7.1-57 §7.1 source ↗

Mixed Practice

In the following exercises, factor.

$x^{2} + x y + 5 x + 5 y$

Answer

(x + y) (x + 5)

7.1-59 §7.1 source ↗

Mixed Practice

In the following exercises, factor.

Area of a rectangle The area of a rectangle with length 6 less than the width is given by the expression $w^{2} - 6 w$ , where $w =$ width. Factor the greatest common factor from the polynomial.

Answer

w (w - 6)

7.1-61 §7.1 source ↗

Mixed Practice

In the following exercises, factor.

The greatest common factor of 36 and 60 is 12. Explain what this means.

Answer

Answers will vary.

Be Prepared 7.4 §7.2 source ↗

Multiply:

(x + 4) (x + 5) .

If you missed this problem, review Example 6.38.

Example 6.38 in the book ↗

Answer

x^{2} + 9 x + 20

Be Prepared 7.5 §7.2 source ↗

Simplify: ⓐ

−9 + (−6)

ⓑ

−9 + 6 .

If you missed this problem, review Example 1.37.

Example 1.37 in the book ↗

Answer

ⓐ −15 ⓑ −3

Be Prepared 7.6 §7.2 source ↗

Simplify: ⓐ

−9 (6)

ⓑ

−9 (−6) .

If you missed this problem, review Example 1.46.

Example 1.46 in the book ↗

Answer

ⓐ −54 ⓑ 54

Try It 7.33 §7.2 source ↗

Factor:

x^{2} + 6 x + 8

.

Answer

(x + 2) (x + 4)

Try It 7.34 §7.2 source ↗

Factor:

y^{2} + 8 y + 15

.

Answer

(y + 3) (y + 5)

Try It 7.35 §7.2 source ↗

Factor:

q^{2} + 10 q + 24

.

Answer

(q + 4) (q + 6)

Try It 7.36 §7.2 source ↗

Factor:

t^{2} + 14 t + 24

.

Answer

(t + 2) (t + 12)

Try It 7.37 §7.2 source ↗

Factor:

x^{2} + 19 x + 60

.

Answer

(x + 4) (x + 15)

Try It 7.38 §7.2 source ↗

Factor:

v^{2} + 23 v + 60

.

Answer

(v + 3) (v + 20)

Try It 7.39 §7.2 source ↗

Factor:

u^{2} - 9 u + 18

.

Answer

(u - 3) (u - 6)

Try It 7.40 §7.2 source ↗

Factor:

y^{2} - 16 y + 63

.

Answer

(y - 7) (y - 9)

Try It 7.41 §7.2 source ↗

Factor:

h^{2} + 4 h - 12

.

Answer

(h - 2) (h + 6)

Try It 7.42 §7.2 source ↗

Factor:

k^{2} + k - 20

.

Answer

(k - 4) (k + 5)

Try It 7.43 §7.2 source ↗

Factor:

x^{2} - 4 x - 12

.

Answer

(x + 2) (x - 6)

Try It 7.44 §7.2 source ↗

Factor:

y^{2} - y - 20

.

Answer

(y + 4) (y - 5)

Try It 7.45 §7.2 source ↗

Factor:

r^{2} - 3 r - 40

.

Answer

(r + 5) (r - 8)

Try It 7.46 §7.2 source ↗

Factor:

s^{2} - 3 s - 10

.

Answer

(s + 2) (s - 5)

Try It 7.47 §7.2 source ↗

Factor:

m^{2} + 4 m + 18

.

Answer

prime

Try It 7.48 §7.2 source ↗

Factor:

n^{2} - 10 n + 12

.

Answer

prime

Try It 7.49 §7.2 source ↗

Factor:

9 m + m^{2} + 18

.

Answer

(m + 3) (m + 6)

Try It 7.50 §7.2 source ↗

Factor:

−7 n + 12 + n^{2}

.

Answer

(n - 3) (n - 4)

Try It 7.51 §7.2 source ↗

Factor:

u^{2} +

11 u v +

28 v^{2}

.

Answer

(u + 4 v) (u + 7 v)

Try It 7.52 §7.2 source ↗

Factor:

x^{2} +

13 x y +

42 y^{2}

.

Answer

(x + 6 y) (x + 7 y)

Try It 7.53 §7.2 source ↗

Factor:

a^{2} -

11 a b +

10 b^{2}

.

Answer

(a - b) (a - 10 b)

Try It 7.54 §7.2 source ↗

Factor:

m^{2} -

13 m n +

12 n^{2}

.

Answer

(m - n) (m - 12 n)

Try It 7.55 §7.2 source ↗

Factor:

x^{2} - 7 x y - 10 y^{2}

.

Answer

prime

Try It 7.56 §7.2 source ↗

Factor:

p^{2} +

15 p q +

20 q^{2}

.

Answer

prime

7.2-63 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$x^{2} + 4 x + 3$

Answer

(x + 1) (x + 3)

7.2-65 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$m^{2} + 12 m + 11$

Answer

(m + 1) (m + 11)

7.2-67 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$a^{2} + 9 a + 20$

Answer

(a + 4) (a + 5)

7.2-69 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$p^{2} + 11 p + 30$

Answer

(p + 5) (p + 6)

7.2-71 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$n^{2} + 19 n + 48$

Answer

(n + 3) (n + 16)

7.2-73 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$a^{2} + 25 a + 100$

Answer

(a + 5) (a + 20)

7.2-75 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$x^{2} - 8 x + 12$

Answer

(x - 2) (x - 6)

7.2-77 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$y^{2} - 18 y + 45$

Answer

(y - 3) (y - 15)

7.2-79 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$x^{2} - 8 x + 7$

Answer

(x - 1) (x - 7)

7.2-81 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$p^{2} + 5 p - 6$

Answer

(p - 1) (p + 6)

7.2-83 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$y^{2} - 6 y - 7$

Answer

(y + 1) (y - 7)

7.2-85 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$x^{2} - x - 12$

Answer

(x - 4) (x + 3)

7.2-87 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$a^{2} - 3 a - 28$

Answer

(a - 7) (a + 4)

7.2-89 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$w^{2} - 5 w - 36$

Answer

(w - 9) (w + 4)

7.2-91 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$x^{2} + x + 5$

Answer

prime

7.2-93 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$8 - 6 x + x^{2}$

Answer

(x - 4) (x - 2)

7.2-95 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$x^{2} - 12 - 11 x$

Answer

(x - 12) (x + 1)

7.2-97 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x y + c y^{2}$

In the following exercises, factor each trinomial of the form $x^{2} + b x y + c y^{2}$ . If the trinomial cannot be factored, answer “Prime.”

$p^{2} + 3 p q + 2 q^{2}$

Answer

(p + q) (p + 2 q)

7.2-99 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x y + c y^{2}$

In the following exercises, factor each trinomial of the form $x^{2} + b x y + c y^{2}$ . If the trinomial cannot be factored, answer “Prime.”

$r^{2} +$ $15 r s +$ $36 s^{2}$

Answer

(r + 3 s) (r + 12 s)

7.2-101 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x y + c y^{2}$

In the following exercises, factor each trinomial of the form $x^{2} + b x y + c y^{2}$ . If the trinomial cannot be factored, answer “Prime.”

$m^{2} -$ $12 m n +$ $20 n^{2}$

Answer

(m - 2 n) (m - 10 n)

7.2-103 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x y + c y^{2}$

In the following exercises, factor each trinomial of the form $x^{2} + b x y + c y^{2}$ . If the trinomial cannot be factored, answer “Prime.”

$x^{2} - 2 x y - 80 y^{2}$

Answer

(x + 8 y) (x - 10 y)

7.2-105 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x y + c y^{2}$

In the following exercises, factor each trinomial of the form $x^{2} + b x y + c y^{2}$ . If the trinomial cannot be factored, answer “Prime.”

$m^{2} -$ $64 m n -$ $65 n^{2}$

Answer

(m + n) (m - 65 n)

7.2-107 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x y + c y^{2}$

In the following exercises, factor each trinomial of the form $x^{2} + b x y + c y^{2}$ . If the trinomial cannot be factored, answer “Prime.”

$a^{2} + 5 a b - 24 b^{2}$

Answer

(a + 8 b) (a - 3 b)

7.2-109 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x y + c y^{2}$

In the following exercises, factor each trinomial of the form $x^{2} + b x y + c y^{2}$ . If the trinomial cannot be factored, answer “Prime.”

$x^{2} - 3 x y - 14 y^{2}$

Answer

prime

7.2-111 §7.2 source ↗

Factor Trinomials of the Form $x^{2} + b x y + c y^{2}$

In the following exercises, factor each trinomial of the form $x^{2} + b x y + c y^{2}$ . If the trinomial cannot be factored, answer “Prime.”

$m^{2} - 5 m n + 30 n^{2}$

Answer

prime

7.2-113 §7.2 source ↗

Mixed Practice

In the following exercises, factor each expression.

$u^{2} - 12 u + 36$

Answer

(u - 6) (u - 6)

7.2-115 §7.2 source ↗

Mixed Practice

In the following exercises, factor each expression.

$x^{2} - 14 x - 32$

Answer

(x + 2) (x - 16)

7.2-117 §7.2 source ↗

Mixed Practice

In the following exercises, factor each expression.

$r^{2} -$ $20 r s +$ $64 s^{2}$

Answer

(r - 4 s) (r - 16 s)

7.2-119 §7.2 source ↗

Mixed Practice

In the following exercises, factor each expression.

$k^{2} + 34 k + 120$

Answer

(k + 4) (k + 30)

7.2-121 §7.2 source ↗

Mixed Practice

In the following exercises, factor each expression.

$y^{2} + 10 y + 15$

Answer

prime

7.2-123 §7.2 source ↗

Mixed Practice

In the following exercises, factor each expression.

$m^{2} + m n - 56 n^{2}$

Answer

(m + 8 n) (m - 7 n)

7.2-125 §7.2 source ↗

Mixed Practice

In the following exercises, factor each expression.

$u^{2} -$ $17 u v +$ $30 v^{2}$

Answer

(u - 15 v) (u - 2 v)

7.2-127 §7.2 source ↗

Mixed Practice

In the following exercises, factor each expression.

$c^{2} - 8 c d + 26 d^{2}$

Answer

prime

7.2-129 §7.2 source ↗

Mixed Practice

In the following exercises, factor each expression.

Consecutive integers Deirdre is thinking of two consecutive integers whose product is 56. The trinomial $x^{2} + x - 56$ describes how these numbers are related. Factor the trinomial.

Answer

(x + 8) (x - 7)

7.2-131 §7.2 source ↗

Mixed Practice

In the following exercises, factor each expression.

Many trinomials of the form $x^{2} + b x + c$ factor into the product of two binomials $(x + m) (x + n)$ . Explain how you find the values of m and n.

Answer

Answers may vary

7.2-133 §7.2 source ↗

Mixed Practice

In the following exercises, factor each expression.

Will factored $x^{2} - x - 20$ as $(x + 5) (x - 4)$ . Bill factored it as $(x + 4) (x - 5)$ . Phil factored it as $(x - 5) (x - 4)$ . Who is correct? Explain why the other two are wrong.

Answer

Answers may vary

Be Prepared 7.8 §7.3 source ↗

Find the GCF of

45 p^{2} and 30 p^{6}

.
If you missed this problem, review Example 7.2.

Example 7.2 in the book ↗

Answer

15 p^{2}

Be Prepared 7.9 §7.3 source ↗

Multiply

(3 y + 4) (2 y + 5)

.
If you missed this problem, review Example 6.40.

Example 6.40 in the book ↗

Answer

6 y^{2} + 23 y + 20

Be Prepared 7.10 §7.3 source ↗

Combine like terms

12 x^{2} +

3 x +

5 x +

9

.
If you missed this problem, review Example 1.24.

Example 1.24 in the book ↗

Answer

12 x^{2} + 8 x + 9

Try It 7.57 §7.3 source ↗

Identify the best method to use to factor each polynomial:

ⓐ $4 y^{2} + 32$

ⓑ $y^{2} + 10 y + 21$

Answer

ⓐ no method ⓑ undo using FOIL ⓒ factor with grouping

Try It 7.58 §7.3 source ↗

Identify the best method to use to factor each polynomial:

ⓐ $a b + a + 4 b + 4$

ⓑ $3 k^{2} + 15$

Answer

ⓐ factor using grouping ⓑ no method ⓒ undo using FOIL

Try It 7.59 §7.3 source ↗

Factor completely:

4 m^{2} - 4 m - 8

.

Answer

4 (m + 1) (m - 2)

Try It 7.60 §7.3 source ↗

Factor completely:

5 k^{2} - 15 k - 50

.

Answer

5 (k + 2) (k - 5)

Try It 7.61 §7.3 source ↗

Factor completely:

3 r^{2} - 9 r + 6

.

Answer

3 (r - 1) (r - 2)

Try It 7.62 §7.3 source ↗

Factor completely:

2 t^{2} - 10 t + 12

.

Answer

2 (t - 2) (t - 3)

Try It 7.63 §7.3 source ↗

Factor completely:

5 x^{3} +

15 x^{2} -

20 x

.

Answer

5 x (x - 1) (x + 4)

Try It 7.64 §7.3 source ↗

Factor completely:

6 y^{3} +

18 y^{2} -

60 y

.

Answer

6 y (y - 2) (y + 5)

Try It 7.65 §7.3 source ↗

Factor completely:

2 a^{2} + 5 a + 3

.

Answer

(a + 1) (2 a + 3)

Try It 7.66 §7.3 source ↗

Factor completely:

4 b^{2} + 5 b + 1

.

Answer

(b + 1) (4 b + 1)

Try It 7.67 §7.3 source ↗

Factor completely:

8 x^{2} - 14 x + 3

.

Answer

(2 x - 3) (4 x - 1)

Try It 7.68 §7.3 source ↗

Factor completely:

10 y^{2} - 37 y + 7

.

Answer

(2 y - 7) (5 y - 1)

Try It 7.69 §7.3 source ↗

Factor completely:

8 a^{2} - 3 a - 5

.

Answer

(a - 1) (8 a + 5)

Try It 7.70 §7.3 source ↗

Factor completely:

6 b^{2} - b - 15

.

Answer

(2 b + 3) (3 b - 5)

Try It 7.71 §7.3 source ↗

Factor completely:

18 x^{2} - 3 x - 10

.

Answer

(3 x + 2) (6 x - 5)

Try It 7.72 §7.3 source ↗

Factor completely:

30 y^{2} - 53 y - 21

.

Answer

(3 y + 1) (10 y - 21)

Try It 7.73 §7.3 source ↗

Factor completely:

15 n^{3} -

85 n^{2} +

100 n

.

Answer

5 n (n - 4) (3 n - 5)

Try It 7.74 §7.3 source ↗

Factor completely:

56 q^{3} +

320 q^{2} -

96 q

.

Answer

8 q (q + 6) (7 q - 2)

Try It 7.75 §7.3 source ↗

Factor:

6 x^{2} + 13 x + 2

.

Answer

(x + 2) (6 x + 1)

Try It 7.76 §7.3 source ↗

Factor:

4 y^{2} + 8 y + 3

.

Answer

(2 y + 1) (2 y + 3)

Try It 7.77 §7.3 source ↗

Factor:

20 h^{2} + 13 h - 15

.

Answer

(4 h + 5) (5 h - 3)

Try It 7.78 §7.3 source ↗

Factor:

6 g^{2} + 19 g - 20

.

Answer

(g + 4) (6 g - 5)

Try It 7.79 §7.3 source ↗

Factor:

10 t^{2} + 19 t - 15

.

Answer

(2 t + 5) (5 t - 3)

Try It 7.80 §7.3 source ↗

Factor:

3 u^{2} + 8 u + 5

.

Answer

(u + 1) (3 u + 5)

Try It 7.81 §7.3 source ↗

Factor:

16 x^{2} - 32 x + 12

.

Answer

4 (2 x - 3) (2 x - 1)

Try It 7.82 §7.3 source ↗

Factor:

18 w^{2} - 39 w + 18

.

Answer

3 (3 w - 2) (2 w - 3)

7.3-135 §7.3 source ↗

Recognize a Preliminary Strategy to Factor Polynomials Completely

In the following exercises, identify the best method to use to factor each polynomial.

ⓐ $10 q^{2} + 50$

ⓑ $a^{2} - 5 a - 14$

Answer

ⓐ factor the GCF, binomial ⓑ Undo FOIL ⓒ factor by grouping

7.3-137 §7.3 source ↗

Recognize a Preliminary Strategy to Factor Polynomials Completely

In the following exercises, identify the best method to use to factor each polynomial.

ⓐ $x^{2} + 4 x - 21$

ⓑ $a b +$ $10 b +$ $4 a +$ $40$

Answer

ⓐ undo FOIL ⓑ factor by grouping ⓒ factor the GCF, binomial

7.3-139 §7.3 source ↗

Factor Trinomials of the form $a x^{2} + b x + c$ with a GCF

In the following exercises, factor completely.

$5 x^{2} + 35 x + 30$

Answer

5 (x + 1) (x + 6)

7.3-141 §7.3 source ↗

Factor Trinomials of the form $a x^{2} + b x + c$ with a GCF

In the following exercises, factor completely.

$2 z^{2} - 2 z - 24$

Answer

2 (z - 4) (z + 3)

7.3-143 §7.3 source ↗

Factor Trinomials of the form $a x^{2} + b x + c$ with a GCF

In the following exercises, factor completely.

$7 v^{2} - 63 v + 56$

Answer

7 (v - 1) (v - 8)

7.3-145 §7.3 source ↗

Factor Trinomials of the form $a x^{2} + b x + c$ with a GCF

In the following exercises, factor completely.

$p^{3} - 8 p^{2} - 20 p$

Answer

p (p - 10) (p + 2)

7.3-147 §7.3 source ↗

Factor Trinomials of the form $a x^{2} + b x + c$ with a GCF

In the following exercises, factor completely.

$3 m^{3} -$ $21 m^{2} +$ $30 m$

Answer

3 m (m - 5) (m - 2)

7.3-149 §7.3 source ↗

Factor Trinomials of the form $a x^{2} + b x + c$ with a GCF

In the following exercises, factor completely.

$5 x^{4} +$ $10 x^{3} -$ $75 x^{2}$

Answer

5 x^{2} (x - 3) (x + 5)

7.3-151 §7.3 source ↗

Factor Trinomials Using Trial and Error

In the following exercises, factor.

$2 t^{2} + 7 t + 5$

Answer

(2 t + 5) (t + 1)

7.3-153 §7.3 source ↗

Factor Trinomials Using Trial and Error

In the following exercises, factor.

$11 x^{2} + 34 x + 3$

Answer

(11 x + 1) (x + 3)

7.3-155 §7.3 source ↗

Factor Trinomials Using Trial and Error

In the following exercises, factor.

$4 w^{2} - 5 w + 1$

Answer

(4 w - 1) (w - 1)

7.3-157 §7.3 source ↗

Factor Trinomials Using Trial and Error

In the following exercises, factor.

$6 p^{2} - 19 p + 10$

Answer

(3 p - 2) (2 p - 5)

7.3-159 §7.3 source ↗

Factor Trinomials Using Trial and Error

In the following exercises, factor.

$4 q^{2} - 7 q - 2$

Answer

(4 q + 1) (q - 2)

7.3-161 §7.3 source ↗

Factor Trinomials Using Trial and Error

In the following exercises, factor.

$4 p^{2} + 17 p - 15$

Answer

(4 p - 3) (p + 5)

7.3-163 §7.3 source ↗

Factor Trinomials Using Trial and Error

In the following exercises, factor.

$16 x^{2} - 32 x + 16$

Answer

16 (x - 1) (x - 1)

7.3-165 §7.3 source ↗

Factor Trinomials Using Trial and Error

In the following exercises, factor.

$30 q^{3} +$ $140 q^{2} +$ $80 q$

Answer

10 q (3 q + 2) (q + 4)

7.3-167 §7.3 source ↗

Factor Trinomials using the ‘ac’ Method

In the following exercises, factor.

$5 n^{2} + 21 n + 4$

Answer

(5 n + 1) (n + 4)

7.3-169 §7.3 source ↗

Factor Trinomials using the ‘ac’ Method

In the following exercises, factor.

$9 z^{2} + 15 z + 4$

Answer

(3 z + 1) (3 z + 4)

7.3-171 §7.3 source ↗

Factor Trinomials using the ‘ac’ Method

In the following exercises, factor.

$4 k^{2} - 16 k + 15$

Answer

(2 k - 3) (2 k - 5)

7.3-173 §7.3 source ↗

Factor Trinomials using the ‘ac’ Method

In the following exercises, factor.

$5 s^{2} - 9 s + 4$

Answer

(5 s - 4) (s - 1)

7.3-175 §7.3 source ↗

Factor Trinomials using the ‘ac’ Method

In the following exercises, factor.

$6 y^{2} + y - 15$

Answer

(3 y + 5) (2 y - 3)

7.3-177 §7.3 source ↗

Factor Trinomials using the ‘ac’ Method

In the following exercises, factor.

$2 n^{2} - 27 n - 45$

Answer

(2 n + 3) (n - 15)

7.3-179 §7.3 source ↗

Factor Trinomials using the ‘ac’ Method

In the following exercises, factor.

$3 x^{2} + 5 x + 4$

Answer

prime

7.3-181 §7.3 source ↗

Factor Trinomials using the ‘ac’ Method

In the following exercises, factor.

$60 y^{2} +$ $290 y -$ $50$

Answer

10 (6 y - 1) (y + 5)

7.3-183 §7.3 source ↗

Factor Trinomials using the ‘ac’ Method

In the following exercises, factor.

$48 z^{3} -$ $102 z^{2} -$ $45 z$

Answer

3 z (8 z + 3) (2 z - 5)

7.3-185 §7.3 source ↗

Factor Trinomials using the ‘ac’ Method

In the following exercises, factor.

$16 s^{2} + 40 s + 24$

Answer

8 (2 s + 3) (s + 1)

7.3-187 §7.3 source ↗

Factor Trinomials using the ‘ac’ Method

In the following exercises, factor.

$48 y^{2} + 12 y - 36$

Answer

12 (4 y - 3) (y + 1)

7.3-189 §7.3 source ↗

Mixed Practice

In the following exercises, factor.

$12 y^{2} - 29 y + 14$

Answer

(4 y - 7) (3 y - 2)

7.3-191 §7.3 source ↗

Mixed Practice

In the following exercises, factor.

$a^{2} - a - 20$

Answer

(a - 5) (a + 4)

7.3-193 §7.3 source ↗

Mixed Practice

In the following exercises, factor.

$6 n^{2} + 5 n - 4$

Answer

(2 n - 1) (3 n + 4)

7.3-195 §7.3 source ↗

Mixed Practice

In the following exercises, factor.

$2 p^{2} + 4 p + 3$

Answer

prime

7.3-197 §7.3 source ↗

Mixed Practice

In the following exercises, factor.

$13 z^{2} + 39 z - 26$

Answer

13 (z^{2} + 3 z - 2)

7.3-199 §7.3 source ↗

Mixed Practice

In the following exercises, factor.

$x^{2} + 3 x - 28$

Answer

(x + 7) (x - 4)

7.3-201 §7.3 source ↗

Mixed Practice

In the following exercises, factor.

$3 p^{2} + 21 p$

Answer

3 p (p + 7)

7.3-203 §7.3 source ↗

Mixed Practice

In the following exercises, factor.

$6 r^{2} + 30 r + 36$

Answer

6 (r + 2) (r + 3)

7.3-205 §7.3 source ↗

Mixed Practice

In the following exercises, factor.

$24 n^{2} + 20 n + 4$

Answer

4 (2 n + 1) (3 n + 1)

7.3-207 §7.3 source ↗

Mixed Practice

In the following exercises, factor.

$x^{2} + 2 x - 24$

Answer

(x + 6) (x - 4)

7.3-209 §7.3 source ↗

Mixed Practice

In the following exercises, factor.

Height of a toy rocket The height of a toy rocket launched with an initial speed of 80 feet per second from the balcony of an apartment building is related to the number of seconds, t, since it is launched by the trinomial $−16 t^{2} +$ $80 t +$ $96$ . Completely factor the trinomial.

Answer

−16 (t - 6) (t + 1)

7.3-211 §7.3 source ↗

Mixed Practice

In the following exercises, factor.

List, in order, all the steps you take when using the “ac” method to factor a trinomial of the form $a x^{2} + b x + c .$

Answer

Answers may vary.

7.3-213 §7.3 source ↗

Mixed Practice

In the following exercises, factor.

What are the questions, in order, that you ask yourself as you start to factor a polynomial? What do you need to do as a result of the answer to each question?

Answer

Answers may vary.

Be Prepared 7.11 §7.4 source ↗

Simplify:

{(12 x)}^{2} .

If you missed this problem, review Example 6.23.

Example 6.23 in the book ↗

Answer

144 x^{2}

Be Prepared 7.12 §7.4 source ↗

Multiply:

{(m + 4)}^{2} .

If you missed this problem, review Example 6.47.

Example 6.47 in the book ↗

Answer

m^{2} + 8 m + 16

Be Prepared 7.13 §7.4 source ↗

Multiply:

{(p - 9)}^{2} .

If you missed this problem, review Example 6.48.

Example 6.48 in the book ↗

Answer

p^{2} - 18 p + 81

Be Prepared 7.14 §7.4 source ↗

Multiply:

(k + 3) (k - 3) .

If you missed this problem, review Example 6.52.

Example 6.52 in the book ↗

Answer

k^{2} - 9

Try It 7.83 §7.4 source ↗

Factor:

4 x^{2} + 12 x + 9

.

Answer

{(2 x + 3)}^{2}

Try It 7.84 §7.4 source ↗

Factor:

9 y^{2} + 24 y + 16

.

Answer

{(3 y + 4)}^{2}

Try It 7.85 §7.4 source ↗

Factor:

64 y^{2} - 80 y + 25

.

Answer

{(8 y - 5)}^{2}

Try It 7.86 §7.4 source ↗

Factor:

16 z^{2} - 72 z + 81

.

Answer

{(4 z - 9)}^{2}

Try It 7.87 §7.4 source ↗

Factor:

49 x^{2} +

84 x y +

36 y^{2}

.

Answer

{(7 x + 6 y)}^{2}

Try It 7.88 §7.4 source ↗

Factor:

64 m^{2} +

112 m n +

49 n^{2}

.

Answer

{(8 m + 7 n)}^{2}

Try It 7.89 §7.4 source ↗

Factor:

16 r^{2} +

30 r s +

9 s^{2}

.

Answer

(8 r + 3 s) (2 r + 3 s)

Try It 7.90 §7.4 source ↗

Factor:

9 u^{2} + 87 u + 100

.

Answer

(3 u + 4) (3 u + 25)

Try It 7.91 §7.4 source ↗

Factor:

8 x^{2} y -

24 x y +

18 y

.

Answer

2 y {(2 x - 3)}^{2}

Try It 7.92 §7.4 source ↗

Factor:

27 p^{2} q +

90 p q +

75 q

.

Answer

3 q {(3 p + 5)}^{2}

Try It 7.93 §7.4 source ↗

Factor:

h^{2} - 81

.

Answer

(h - 9) (h + 9)

Try It 7.94 §7.4 source ↗

Factor:

k^{2} - 121

.

Answer

(k - 11) (k + 11)

Try It 7.95 §7.4 source ↗

Factor:

m^{2} - 1

.

Answer

(m - 1) (m + 1)

Try It 7.96 §7.4 source ↗

Factor:

81 y^{2} - 1

.

Answer

(9 y - 1) (9 y + 1)

Try It 7.97 §7.4 source ↗

Factor:

196 m^{2} - 25 n^{2}

.

Answer

(14 m - 5 n) (14 m + 5 n)

Try It 7.98 §7.4 source ↗

Factor:

144 p^{2} - 9 q^{2}

.

Answer

9 (4 p - q) (4 p + q)

Try It 7.99 §7.4 source ↗

Factor:

144 - x^{2}

.

Answer

(12 - x) (12 + x)

Try It 7.100 §7.4 source ↗

Factor:

169 - p^{2}

.

Answer

(13 - p) (13 + p)

Try It 7.101 §7.4 source ↗

Factor:

a^{4} - b^{4}

.

Answer

(a^{2} + b^{2}) (a + b) (a - b)

Try It 7.102 §7.4 source ↗

Factor:

x^{4} - 16

.

Answer

(x^{2} + 4) (x + 2) (x - 2)

Try It 7.103 §7.4 source ↗

Factor:

7 x y^{2} - 175 x

.

Answer

7 x (y - 5) (y + 5)

Try It 7.104 §7.4 source ↗

Factor:

45 a^{2} b - 80 b

.

Answer

5 b (3 a - 4) (3 a + 4)

Try It 7.105 §7.4 source ↗

Factor:

8 a^{2} + 200

.

Answer

8 (a^{2} + 25)

Try It 7.106 §7.4 source ↗

Factor:

36 y^{2} + 81

.

Answer

9 (4 y^{2} + 9)

Try It 7.107 §7.4 source ↗

Factor:

x^{3} + 27

.

Answer

(x + 3) (x^{2} - 3 x + 9)

Try It 7.108 §7.4 source ↗

Factor:

y^{3} + 8

.

Answer

(y + 2) (y^{2} - 2 y + 4)

Try It 7.109 §7.4 source ↗

Factor:

u^{3} - 125

.

Answer

(u - 5) (u^{2} + 5 u + 25)

Try It 7.110 §7.4 source ↗

Factor:

v^{3} - 343

.

Answer

(v - 7) (v^{2} + 7 v + 49)

Try It 7.111 §7.4 source ↗

Factor:

64 - 27 x^{3}

.

Answer

(4 - 3 x) (16 + 12 x + 9 x^{2})

Try It 7.112 §7.4 source ↗

Factor:

27 - 8 y^{3}

.

Answer

(3 - 2 y) (9 + 6 y + 4 y^{2})

Try It 7.113 §7.4 source ↗

Factor:

8 x^{3} - 27 y^{3}

.

Answer

(2 x - 3 y) (4 x^{2} + 6 x y + 9 y^{2})

Try It 7.114 §7.4 source ↗

Factor:

1000 m^{3} -

125 n^{3}

.

Answer

125 (4 m^{2} + 2 m n + n^{2}) (2 m - n)

Try It 7.115 §7.4 source ↗

Factor:

500 p^{3} + 4 q^{3}

.

Answer

4 (5 p + q) (25 p^{2} - 5 p q + q^{2})

Try It 7.116 §7.4 source ↗

Factor:

432 c^{3} + 686 d^{3}

.

Answer

2 (6 c + 7 d) (36 c^{2} - 42 c d + 49 d^{2})

7.4-215 §7.4 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor.

$16 y^{2} + 24 y + 9$

Answer

{(4 y + 3)}^{2}

7.4-217 §7.4 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor.

$36 s^{2} + 84 s + 49$

Answer

{(6 s + 7)}^{2}

7.4-219 §7.4 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor.

$100 x^{2} - 20 x + 1$

Answer

{(10 x - 1)}^{2}

7.4-221 §7.4 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor.

$25 n^{2} -$ $120 n +$ $144$

Answer

{(5 n - 12)}^{2}

7.4-223 §7.4 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor.

$49 x^{2} -$ $28 x y +$ $4 y^{2}$

Answer

{(7 x - 2 y)}^{2}

7.4-225 §7.4 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor.

$25 n^{2} + 25 n + 4$

Answer

(5 n + 4) (5 n + 1)

7.4-227 §7.4 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor.

$64 m^{2} - 16 m + 1$

Answer

(8 m - 1)^{2}

7.4-229 §7.4 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor.

$10 k^{2} +$ $80 k +$ $160$

Answer

10 {(k + 4)}^{2}

7.4-231 §7.4 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor.

$75 u^{3} -$ $30 u^{2} v +$ $3 u v^{2}$

Answer

3 u {(5 u - v)}^{2}

7.4-233 §7.4 source ↗

Factor Differences of Squares

In the following exercises, factor.

$x^{2} - 16$

Answer

(x - 4) (x + 4)

7.4-235 §7.4 source ↗

Factor Differences of Squares

In the following exercises, factor.

$25 v^{2} - 1$

Answer

(5 v - 1) (5 v + 1)

7.4-237 §7.4 source ↗

Factor Differences of Squares

In the following exercises, factor.

$121 x^{2} - 144 y^{2}$

Answer

(11 x - 12 y) (11 x + 12 y)

7.4-239 §7.4 source ↗

Factor Differences of Squares

In the following exercises, factor.

$169 c^{2} - 36 d^{2}$

Answer

(13 c - 6 d) (13 c + 6 d)

7.4-241 §7.4 source ↗

Factor Differences of Squares

In the following exercises, factor.

$4 - 49 x^{2}$

Answer

(2 - 7 x) (2 + 7 x)

7.4-243 §7.4 source ↗

Factor Differences of Squares

In the following exercises, factor.

$16 z^{4} - 1$

Answer

(2 z - 1) (2 z + 1) (4 z^{2} + 1)

7.4-245 §7.4 source ↗

Factor Differences of Squares

In the following exercises, factor.

$5 q^{2} - 45$

Answer

5 (q - 3) (q + 3)

7.4-247 §7.4 source ↗

Factor Differences of Squares

In the following exercises, factor.

$24 p^{2} + 54$

Answer

6 (4 p^{2} + 9)

7.4-249 §7.4 source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor.

$x^{3} + 125$

Answer

(x + 5) (x^{2} - 5 x + 25)

7.4-251 §7.4 source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor.

$z^{3} - 27$

Answer

(z - 3) (z^{2} + 3 z + 9)

7.4-253 §7.4 source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor.

$8 - 343 t^{3}$

Answer

(2 - 7 t) (4 + 14 t + 49 t^{2})

7.4-255 §7.4 source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor.

$8 y^{3} - 125 z^{3}$

Answer

(2 y - 5 z) (4 y^{2} + 10 y z + 25 z^{2})

7.4-257 §7.4 source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor.

$7 k^{3} + 56$

Answer

7 (k + 2) (k^{2} - 2 k + 4)

7.4-259 §7.4 source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor.

$2 - 16 y^{3}$

Answer

2 (1 - 2 y) (1 + 2 y + 4 y^{2})

7.4-261 §7.4 source ↗

Mixed Practice

In the following exercises, factor.

$64 a^{2} - 25$

Answer

(8 a - 5) (8 a + 5)

7.4-263 §7.4 source ↗

Mixed Practice

In the following exercises, factor.

$27 q^{2} - 3$

Answer

3 (3 q - 1) (3 q + 1)

7.4-265 §7.4 source ↗

Mixed Practice

In the following exercises, factor.

$16 x^{2} - 72 x + 81$

Answer

{(4 x - 9)}^{2}

7.4-267 §7.4 source ↗

Mixed Practice

In the following exercises, factor.

$8 p^{2} + 2$

Answer

2 (4 p^{2} + 1)

7.4-269 §7.4 source ↗

Mixed Practice

In the following exercises, factor.

$125 - 8 y^{3}$

Answer

(5 - 2 y) (25 + 10 y + 4 y^{2})

7.4-271 §7.4 source ↗

Mixed Practice

In the following exercises, factor.

$45 n^{2} + 60 n + 20$

Answer

5 {(3 n + 2)}^{2}

7.4-273 §7.4 source ↗

Mixed Practice

In the following exercises, factor.

Landscaping Sue and Alan are planning to put a 15 foot square swimming pool in their backyard. They will surround the pool with a tiled deck, the same width on all sides. If the width of the deck is w, the total area of the pool and deck is given by the trinomial $4 w^{2} + 60 w + 225$ . Factor the trinomial.

Answer

{(2 w + 15)}^{2}

7.4-275 §7.4 source ↗

Mixed Practice

In the following exercises, factor.

Why was it important to practice using the binomial squares pattern in the chapter on multiplying polynomials?

Answer

Answers may vary.

7.4-277 §7.4 source ↗

Mixed Practice

In the following exercises, factor.

Explain why $n^{2} +$ $25 \neq {(n + 5)}^{2}$ . Use algebra, words, or pictures.

Answer

Answers may vary.

Be Prepared 7.15 §7.5 source ↗

Factor

y^{2} - 2 y - 24

.
If you missed this problem, review Example 7.23.

Example 7.23 in the book ↗

Answer

(y - 6) (y + 4)

Be Prepared 7.16 §7.5 source ↗

Factor

3 t^{2} + 17 t + 10

.
If you missed this problem, review Example 7.38.

Example 7.38 in the book ↗

Answer

(3 t + 2) (t + 5)

Be Prepared 7.17 §7.5 source ↗

Factor

36 p^{2} - 60 p + 25

.
If you missed this problem, review Example 7.42.

Example 7.42 in the book ↗

Answer

{(6 p - 5)}^{2}

Be Prepared 7.18 §7.5 source ↗

Factor

5 x^{2} - 80

.
If you missed this problem, review Example 7.52.

Example 7.52 in the book ↗

Answer

5 (x - 4) (x + 4)

Try It 7.117 §7.5 source ↗

Factor completely:

3 a^{4} + 18 a^{3}

.

Answer

3 a^{3} (a + 6)

Try It 7.118 §7.5 source ↗

Factor completely:

45 b^{6} + 27 b^{5}

.

Answer

9 b^{5} (5 b + 3)

Try It 7.119 §7.5 source ↗

Factor completely:

10 a^{2} - 17 a + 6

.

Answer

(5 a - 6) (2 a - 1)

Try It 7.120 §7.5 source ↗

Factor completely:

8 x^{2} - 18 x + 9

.

Answer

(2 x - 3) (4 x - 3)

Try It 7.121 §7.5 source ↗

Factor completely:

x^{3} + 36 x

.

Answer

x (x^{2} + 36)

Try It 7.122 §7.5 source ↗

Factor completely:

27 y^{2} + 48

.

Answer

3 (9 y^{2} + 16)

Try It 7.123 §7.5 source ↗

Factor completely:

16 x^{3} - 36 x

.

Answer

4 x (2 x - 3) (2 x + 3)

Try It 7.124 §7.5 source ↗

Factor completely:

27 y^{2} - 48

.

Answer

3 (3 y - 4) (3 y + 4)

Try It 7.125 §7.5 source ↗

Factor completely:

4 x^{2} +

20 x y +

25 y^{2}

.

Answer

{(2 x + 5 y)}^{2}

Try It 7.126 §7.5 source ↗

Factor completely:

9 m^{2} +

42 m n +

49 n^{2}

.

Answer

{(3 m + 7 n)}^{2}

Try It 7.127 §7.5 source ↗

Factor completely:

8 y^{2} + 16 y - 24

.

Answer

8 (y - 1) (y + 3)

Try It 7.128 §7.5 source ↗

Factor completely:

5 u^{2} - 15 u - 270

.

Answer

5 (u - 9) (u + 6)

Try It 7.129 §7.5 source ↗

Factor completely:

250 m^{3} + 432

.

Answer

2 (5 m + 6) (25 m^{2} - 30 m + 36)

Try It 7.130 §7.5 source ↗

Factor completely:

81 q^{3} + 192

.

Answer

3 (3 q + 4) (9 q^{2} - 12 q + 16)

Try It 7.131 §7.5 source ↗

Factor completely:

4 a^{4} - 64

.

Answer

4 (a^{2} + 4) (a - 2) (a + 2)

Try It 7.132 §7.5 source ↗

Factor completely:

7 y^{4} - 7

.

Answer

7 (y^{2} + 1) (y - 1) (y + 1)

Try It 7.133 §7.5 source ↗

Factor completely:

6 x^{2} -

12 x c +

6 b x -

12 b c

.

Answer

6 (x + b) (x - 2 c)

Try It 7.134 §7.5 source ↗

Factor completely:

16 x^{2} +

24 x y -

4 x -

6 y

.

Answer

2 (4 x - 1) (2 x + 3 y)

Try It 7.135 §7.5 source ↗

Factor completely:

4 p^{2} - 16 p + 12

.

Answer

4 (p - 1) (p - 3)

Try It 7.136 §7.5 source ↗

Factor completely:

6 q^{2} - 9 q - 6

.

Answer

3 (q - 2) (2 q + 1)

7.5-279 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$10 x^{4} + 35 x^{3}$

Answer

5 x^{3} (2 x + 7)

7.5-281 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$y^{2} + 10 y - 39$

Answer

(y - 3) (y + 13)

7.5-283 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$2 n^{2} + 13 n - 7$

Answer

(2 n - 1) (n + 7)

7.5-285 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$a^{5} + 9 a^{3}$

Answer

a^{3} (a^{2} + 9)

7.5-287 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$121 r^{2} - s^{2}$

Answer

(11 r - s) (11 r + s)

7.5-289 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$8 m^{2} - 32$

Answer

8 (m - 2) (m + 2)

7.5-291 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$25 w^{2} - 60 w + 36$

Answer

{(5 w - 6)}^{2}

7.5-293 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$m^{2} +$ $14 m n +$ $49 n^{2}$

Answer

{(m + 7 n)}^{2}

7.5-295 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$7 b^{2} + 7 b - 42$

Answer

7 (b + 3) (b - 2)

7.5-297 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$3 x^{3} - 81$

Answer

3 (x - 3) (x^{2} + 3 x + 9)

7.5-299 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$k^{4} - 16$

Answer

(k - 2) (k + 2) (k^{2} + 4)

7.5-301 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$15 p q -$ $15 p +$ $12 q -$ $12$

Answer

3 (5 p + 4) (q - 1)

7.5-303 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$4 x^{2} + 40 x + 84$

Answer

4 (x + 3) (x + 7)

7.5-305 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$u^{5} + u^{2}$

Answer

u^{2} (u + 1) (u^{2} - u + 1)

7.5-307 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$4 c^{2} +$ $20 c d +$ $81 d^{2}$

Answer

prime

7.5-309 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$10 m^{4} - 6250$

Answer

10 (m - 5) (m + 5) (m^{2} + 25)

7.5-311 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

Watermelon drop A springtime tradition at the University of California San Diego is the Watermelon Drop, where a watermelon is dropped from the seventh story of Urey Hall.

ⓐ The binomial $−16 t^{2} + 80$ gives the height of the watermelon $t$ seconds after it is dropped. Factor the greatest common factor from this binomial.

ⓑ If the watermelon is thrown down with initial velocity 8 feet per second, its height after $t$ seconds is given by the trinomial $−16 t^{2} - 8 t + 80$ . Completely factor this trinomial.

Answer

ⓐ

−16 (t^{2} - 5)

ⓑ

−8 (2 t + 5) (t - 2)

7.5-313 §7.5 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

The difference of squares $y^{4} - 625$ can be factored as $(y^{2} - 25) (y^{2} + 25)$ . But it is not completely factored. What more must be done to completely factor it?

Answer

Answer may vary.

Be Prepared 7.19 §7.6 source ↗

Solve:

5 y - 3 = 0

.
If you missed this problem, review Example 2.27.

Example 2.27 in the book ↗

Answer

y = \frac{3}{5}

Be Prepared 7.20 §7.6 source ↗

Solve:

10 a = 0

.
If you missed this problem, review Example 2.13.

Example 2.13 in the book ↗

Answer

a = 0

Be Prepared 7.21 §7.6 source ↗

Combine like terms:

12 x^{2} - 6 x + 4 x

.
If you missed this problem, review Example 1.24.

Example 1.24 in the book ↗

Answer

12 x^{2} - 2 x

Be Prepared 7.22 §7.6 source ↗

Factor

n^{3} - 9 n^{2} - 22 n

completely.
If you missed this problem, review Example 7.32.

Example 7.32 in the book ↗

Answer

n (n - 11) (n + 2)

Try It 7.137 §7.6 source ↗

Solve:

(x - 3) (x + 5) =

0

.

Answer

x = 3, x = −5

Try It 7.138 §7.6 source ↗

Solve:

(y - 6) (y + 9) =

0

.

Answer

y = 6, y = −9

Try It 7.139 §7.6 source ↗

Solve:

(3 m - 2) (2 m + 1) =

0

.

Answer

m = \frac{2}{3}, m = - \frac{1}{2}

Try It 7.140 §7.6 source ↗

Solve:

(4 p + 3) (4 p - 3) =

0

.

Answer

p = - \frac{3}{4}, p = \frac{3}{4}

Try It 7.141 §7.6 source ↗

Solve:

2 u (5 u - 1) = 0

.

Answer

u = 0, u = \frac{1}{5}

Try It 7.142 §7.6 source ↗

Solve:

w (2 w + 3) = 0

.

Answer

w = 0, w = - \frac{3}{2}

Try It 7.143 §7.6 source ↗

Solve:

{(x + 1)}^{2} = 0

.

Answer

x = −1

Try It 7.144 §7.6 source ↗

Solve:

{(v - 2)}^{2} = 0

.

Answer

v = 2

Try It 7.145 §7.6 source ↗

Solve:

x^{2} - x - 12 = 0

.

Answer

x = 4, x = −3

Try It 7.146 §7.6 source ↗

Solve:

b^{2} + 9 b + 14 = 0

.

Answer

b = −2, b = −7

Try It 7.147 §7.6 source ↗

Solve:

3 c^{2} = 10 c - 8

.

Answer

c = 2, c = \frac{4}{3}

Try It 7.148 §7.6 source ↗

Solve:

2 d^{2} - 5 d = 3

.

Answer

d = 3, d = - \frac{1}{2}

Try It 7.149 §7.6 source ↗

Solve:

6 a^{2} + 9 a = 3 a

.

Answer

a = 0, a = −1

Try It 7.150 §7.6 source ↗

Solve:

45 b^{2} -

2 b =

−17 b

.

Answer

b = 0, b = - \frac{1}{3}

Try It 7.151 §7.6 source ↗

Solve:

25 p^{2} = 49

.

Answer

p = \frac{7}{5}, p = - \frac{7}{5}

Try It 7.152 §7.6 source ↗

Solve:

36 x^{2} = 121

.

Answer

x =

\frac{11}{6},

x =

-

\frac{11}{6}

Try It 7.153 §7.6 source ↗

Solve:

(2 m + 1) (m + 3) =

12 m

.

Answer

m = 1, m = \frac{3}{2}

Try It 7.154 §7.6 source ↗

Solve:

(k + 1) (k - 1) =

8

.

Answer

k = 3, k = −3

Try It 7.155 §7.6 source ↗

Solve:

8 x^{3} =

24 x^{2} -

18 x

.

Answer

x = 0, x = \frac{3}{2}

Try It 7.156 §7.6 source ↗

Solve:

16 y^{2} =

32 y^{3} +

2 y

.

Answer

y = 0, y = \frac{1}{4}

Try It 7.157 §7.6 source ↗

Solve:

18 a^{2} -

30 =

−33 a

.

Answer

a = - \frac{5}{2}, a = \frac{2}{3}

Try It 7.158 §7.6 source ↗

Solve:

123 b =

−6 -

60 b^{2}

.

Answer

b = −2, b = - \frac{1}{20}

Try It 7.159 §7.6 source ↗

The product of two consecutive integers is

240

. Find the integers.

Answer

−15,

−16 and 15,

16

Try It 7.160 §7.6 source ↗

The product of two consecutive integers is

420

. Find the integers.

Answer

−21,

−20 and 20,

21

Try It 7.161 §7.6 source ↗

A rectangular sign has an area of 30 square feet. The length of the sign is one foot more than the width. Find the length and width of the sign.

Answer

5

feet and

6

feet

Try It 7.162 §7.6 source ↗

A rectangular patio has an area of 180 square feet. The width of the patio is three feet less than the length. Find the length and width of the patio.

Answer

12

feet and

15

feet

Try It 7.163 §7.6 source ↗

A boat’s sail is a right triangle. The length of one side of the sail is 7 feet more than the other side. The hypotenuse is 13. Find the lengths of the two sides of the sail.

Answer

5

feet and

12

feet

Try It 7.164 §7.6 source ↗

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg.

Answer

24

feet and

25

feet

7.6-315 §7.6 source ↗

Use the Zero Product Property

In the following exercises, solve.

$(x - 3) (x + 7) =$ $0$

Answer

x = 3, x = −7

7.6-317 §7.6 source ↗

Use the Zero Product Property

In the following exercises, solve.

$(3 a - 10) (2 a - 7) =$ $0$

Answer

a =

10 / 3,

a =

7 / 2

7.6-319 §7.6 source ↗

Use the Zero Product Property

In the following exercises, solve.

$6 m (12 m - 5) = 0$

Answer

m = 0, m = 5 / 12

7.6-321 §7.6 source ↗

Use the Zero Product Property

In the following exercises, solve.

${(y - 3)}^{2} = 0$

Answer

y = 3

7.6-323 §7.6 source ↗

Use the Zero Product Property

In the following exercises, solve.

${(2 x - 1)}^{2} = 0$

Answer

x = 1 / 2

7.6-325 §7.6 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$x^{2} + 7 x + 12 = 0$

Answer

x = −3, x = −4

7.6-327 §7.6 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$5 a^{2} - 26 a = 24$

Answer

a = −4 / 5, a = 6

7.6-329 §7.6 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$4 m^{2} = 17 m - 15$

Answer

m = 5 / 4, m = 3

7.6-331 §7.6 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$7 a^{2} + 14 a = 7 a$

Answer

a = −1, a = 0

7.6-333 §7.6 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$49 m^{2} = 144$

Answer

m =

12 / 7,

m =

−12 / 7

7.6-335 §7.6 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$(y - 3) (y + 2) =$ $4 y$

Answer

y = −1, y = 6

7.6-337 §7.6 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$(2 x + 1) (x - 3) =$ $−4 x$

Answer

x = 3 / 2, x = −1

7.6-339 §7.6 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$16 p^{3} =$ $24 p^{2} - 9 p$

Answer

p = 0, p = ¾

7.6-341 §7.6 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$20 x^{2} -$ $60 x =$ $−45$

Answer

x = 3 / 2

7.6-343 §7.6 source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

The product of two consecutive integers is 56. Find the integers.

Answer

7 and 8;

-

8 and −7

7.6-345 §7.6 source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

The area of a rectangular carpet is 28 square feet. The length is three feet more than the width. Find the length and the width of the carpet.

Answer

4 feet and 7 feet

7.6-347 §7.6 source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

A pennant is shaped like a right triangle, with hypotenuse 10 feet. The length of one side of the pennant is two feet longer than the length of the other side. Find the length of the two sides of the pennant.

Answer

6 feet and 8 feet

7.6-349 §7.6 source ↗

Mixed Practice

In the following exercises, solve.

$(x + 8) (x - 3) =$ $0$

Answer

x = −8, x = 3

7.6-351 §7.6 source ↗

Mixed Practice

In the following exercises, solve.

$p^{2} + 12 p + 11 = 0$

Answer

p = −1, p = −11

7.6-353 §7.6 source ↗

Mixed Practice

In the following exercises, solve.

$m^{2} = 6 m + 16$

Answer

m = −2, m = 8

7.6-355 §7.6 source ↗

Mixed Practice

In the following exercises, solve.

$a^{3} - a^{2} - 42 a = 0$

Answer

a =

0,

a =

−6,

a =

7

7.6-357 §7.6 source ↗

Mixed Practice

In the following exercises, solve.

The product of two consecutive integers is 110. Find the integers.

Answer

10 and 11;

-

11 and −10

7.6-359 §7.6 source ↗

Mixed Practice

In the following exercises, solve.

Area of a patio If each side of a square patio is increased by 4 feet, the area of the patio would be 196 square feet. Solve the equation ${(s + 4)}^{2} = 196$ for s to find the length of a side of the patio.

Answer

10 feet

7.6-361 §7.6 source ↗

Mixed Practice

In the following exercises, solve.

Explain how you solve a quadratic equation. How many answers do you expect to get for a quadratic equation?

Answer

Answers may vary.

7.R-363 §7.R source ↗

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$42, 60$

Answer

6

7.R-365 §7.R source ↗

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$90, 150, 105$

Answer

15

7.R-367 §7.R source ↗

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$24 x - 42$

Answer

6 (4 x - 7)

7.R-369 §7.R source ↗

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$15 m^{4} + 6 m^{2} n$

Answer

3 m^{2} (5 m^{2} + 2 n)

7.R-371 §7.R source ↗

Factor by Grouping

In the following exercises, factor by grouping.

$a x - a y + b x - b y$

Answer

(a + b) (x - y)

7.R-373 §7.R source ↗

Factor by Grouping

In the following exercises, factor by grouping.

$x^{2} + 7 x - 3 x - 21$

Answer

(x - 3) (x + 7)

7.R-375 §7.R source ↗

Factor by Grouping

In the following exercises, factor by grouping.

$m^{3} + m^{2} + m + 1$

Answer

(m^{2} + 1) (m + 1)

7.R-377 §7.R source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$u^{2} + 17 u + 72$

Answer

(u + 8) (u + 9)

7.R-379 §7.R source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$k^{2} - 16 k + 60$

Answer

(k - 6) (k - 10)

7.R-381 §7.R source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$y^{2} + 6 y - 7$

Answer

(y + 7) (y - 1)

7.R-383 §7.R source ↗

Factor Trinomials of the Form $x^{2} + b x + c$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$s^{2} - 2 s - 8$

Answer

(s - 4) (s + 2)

7.R-385 §7.R source ↗

Factor Trinomials of the Form $x^{2} + b x y + c y^{2}$

In the following examples, factor each trinomial of the form $x^{2} + b x y + c y^{2}$ .

$x^{2} +$ $12 x y +$ $35 y^{2}$

Answer

(x + 5 y) (x + 7 y)

7.R-387 §7.R source ↗

Factor Trinomials of the Form $x^{2} + b x y + c y^{2}$

In the following examples, factor each trinomial of the form $x^{2} + b x y + c y^{2}$ .

$a^{2} + 4 a b - 21 b^{2}$

Answer

(a + 7 b) (a - 3 b)

7.R-389 §7.R source ↗

Recognize a Preliminary Strategy to Factor Polynomials Completely

In the following exercises, identify the best method to use to factor each polynomial.

$y^{2} - 17 y + 42$

Answer

Undo FOIL

7.R-391 §7.R source ↗

Recognize a Preliminary Strategy to Factor Polynomials Completely

In the following exercises, identify the best method to use to factor each polynomial.

$8 a^{3} + 72 a$

Answer

Factor the GCF

7.R-393 §7.R source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ with a GCF

In the following exercises, factor completely.

$6 x^{2} + 42 x + 60$

Answer

6 (x + 2) (x + 5)

7.R-395 §7.R source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ with a GCF

In the following exercises, factor completely.

$3 n^{4} -$ $12 n^{3} -$ $96 n^{2}$

Answer

3 n^{2} (n - 8) (n + 4)

7.R-397 §7.R source ↗

Factor Trinomials Using the “ac” Method

In the following exercises, factor.

$2 x^{2} + 9 x + 4$

Answer

(x + 4) (2 x + 1)

7.R-399 §7.R source ↗

Factor Trinomials Using the “ac” Method

In the following exercises, factor.

$18 a^{2} - 9 a + 1$

Answer

(3 a - 1) (6 a - 1)

7.R-401 §7.R source ↗

Factor Trinomials Using the “ac” Method

In the following exercises, factor.

$15 p^{2} + 2 p - 8$

Answer

(5 p + 4) (3 p - 2)

7.R-403 §7.R source ↗

Factor Trinomials Using the “ac” Method

In the following exercises, factor.

$40 s^{2} - s - 6$

Answer

(5 s - 2) (8 s + 3)

7.R-405 §7.R source ↗

Factor Trinomials with a GCF Using the “ac” Method

In the following exercises, factor.

$3 x^{2} + 3 x - 36$

Answer

3 (x + 4) (x - 3)

7.R-407 §7.R source ↗

Factor Trinomials with a GCF Using the “ac” Method

In the following exercises, factor.

$60 y^{2} - 85 y - 25$

Answer

5 (4 y + 1) (3 y - 5)

7.R-409 §7.R source ↗

Factor Perfect Square Trinomials

In the following exercises, factor.

$25 x^{2} + 30 x + 9$

Answer

{(5 x + 3)}^{2}

7.R-411 §7.R source ↗

Factor Perfect Square Trinomials

In the following exercises, factor.

$36 a^{2} -$ $84 a b +$ $49 b^{2}$

Answer

{(6 a - 7 b)}^{2}

7.R-413 §7.R source ↗

Factor Perfect Square Trinomials

In the following exercises, factor.

$40 x^{2} +$ $360 x +$ $810$

Answer

10 {(2 x + 9)}^{2}

7.R-415 §7.R source ↗

Factor Perfect Square Trinomials

In the following exercises, factor.

$2 y^{3} -$ $16 y^{2} +$ $32 y$

Answer

2 y {(y - 4)}^{2}

7.R-417 §7.R source ↗

Factor Differences of Squares

In the following exercises, factor.

$81 r^{2} - 25$

Answer

(9 r - 5) (9 r + 5)

7.R-419 §7.R source ↗

Factor Differences of Squares

In the following exercises, factor.

$169 m^{2} - n^{2}$

Answer

(13 m + n) (13 m - n)

7.R-421 §7.R source ↗

Factor Differences of Squares

In the following exercises, factor.

$25 p^{2} - 1$

Answer

(5 p - 1) (5 p + 1)

7.R-423 §7.R source ↗

Factor Differences of Squares

In the following exercises, factor.

$9 - 121 y^{2}$

Answer

(3 + 11 y) (3 - 11 y)

7.R-425 §7.R source ↗

Factor Differences of Squares

In the following exercises, factor.

$20 x^{2} - 125$

Answer

5 (2 x - 5) (2 x + 5)

7.R-427 §7.R source ↗

Factor Differences of Squares

In the following exercises, factor.

$49 u^{3} - 9 u$

Answer

u (7 u + 3) (7 u - 3)

7.R-429 §7.R source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor.

$a^{3} - 125$

Answer

(a - 5) (a^{2} + 5 a + 25)

7.R-431 §7.R source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor.

$2 m^{3} + 54$

Answer

2 (m + 3) (m^{2} - 3 m + 9)

7.R-433 §7.R source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$24 x^{3} + 44 x^{2}$

Answer

4 x^{2} (6 x + 11)

7.R-435 §7.R source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$16 n^{2} -$ $56 m n +$ $49 m^{2}$

Answer

{(4 n - 7 m)}^{2}

7.R-437 §7.R source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$5 r^{2} + 22 r - 48$

Answer

(r + 6) (5 r - 8)

7.R-439 §7.R source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$n^{4} - 81$

Answer

(n^{2} + 9) (n + 3) (n - 3)

7.R-441 §7.R source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$5 x^{2} + 5 x - 60$

Answer

5 (x - 3) (x + 4)

7.R-443 §7.R source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$m^{3} + 125$

Answer

(m + 5) (m^{2} - 5 m + 25)

7.R-445 §7.R source ↗

Use the Zero Product Property

In the following exercises, solve.

$(a - 3) (a + 7) =$ $0$

Answer

a = 3, a = −7

7.R-447 §7.R source ↗

Use the Zero Product Property

In the following exercises, solve.

$3 m (2 m - 5) (m + 6) =$ $0$

Answer

m = 0 m = –6 m = \frac{5}{2}

7.R-449 §7.R source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$x^{2} + 9 x + 20 = 0$

Answer

x = −4, x = −5

7.R-451 §7.R source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$2 p^{2} - 11 p = 40$

Answer

p = - \frac{5}{2}, p = 8

7.R-453 §7.R source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$144 m^{2} - 25 = 0$

Answer

m =

\frac{5}{12},

m =

-

\frac{5}{12}

7.R-455 §7.R source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

The product of two consecutive numbers is $462$ . Find the numbers.

Answer

−21 and −22;

21 and 22

7.T-457 §7.T source ↗

In the following exercises, find the Greatest Common Factor in each expression.

$14 y - 42$

Answer

14 (y - 3)

7.T-459 §7.T source ↗

In the following exercises, find the Greatest Common Factor in each expression.

$80 a^{2} + 120 a^{3}$

Answer

40 a^{2} (2 + 3 a)

7.T-461 §7.T source ↗

In the following exercises, factor completely.

$x^{2} + 13 x + 42$

Answer

(x + 7) (x + 6)

7.T-463 §7.T source ↗

In the following exercises, factor completely.

$3 a^{3} - 6 a^{2} - 72 a$

Answer

3 a (a - 6) (a + 4)

7.T-465 §7.T source ↗

In the following exercises, factor completely.

$5 n^{2} + 30 n + 45$

Answer

5 (n + 3)^{2}

7.T-467 §7.T source ↗

In the following exercises, factor completely.

$x y - 8 y + 7 x - 56$

Answer

(x - 8) (y + 7)

7.T-469 §7.T source ↗

In the following exercises, factor completely.

$9 s^{2} - 12 s + 4$

Answer

{(3 s - 2)}^{2}

7.T-471 §7.T source ↗

In the following exercises, factor completely.

$100 - a^{2}$

Answer

(10 - a) (10 + a)

7.T-473 §7.T source ↗

In the following exercises, factor completely.

$3 x^{2} - 75 y^{2}$

Answer

3 (x + 5 y) (x - 5 y)

7.T-475 §7.T source ↗

In the following exercises, factor completely.

$a b - 3 b - 2 a + 6$

Answer

(a - 3) (b - 2)

7.T-477 §7.T source ↗

In the following exercises, factor completely.

$8 m^{2} + 22 m + 5$

Answer

(4 m + 1) (2 m + 5)

7.T-479 §7.T source ↗

In the following exercises, solve.

$y^{2} = y + 132$

Answer

y = −11, y = 12

7.T-481 §7.T source ↗

In the following exercises, solve.

$9 b^{2} - 9 = 0$

Answer

b = 1, b = −1

7.T-483 §7.T source ↗

In the following exercises, solve.

$4 n^{2} +$ $19 n +$ $21 =$ $0$

Answer

n = - \frac{7}{4}, n = −3

7.T-485 §7.T source ↗

In the following exercises, solve.

The product of two consecutive integers is 156. Find the integers.

Answer

12 and 13;

-

13 and −12