Content check — Intermediate Algebra, Chapter 6: Factoring

Be Prepared 6.1 §6.1 source ↗

Factor 56 into primes.
If you missed this problem, review Example 1.2.

Answer

2 \cdot 2 \cdot 2 \cdot 7

Be Prepared 6.2 §6.1 source ↗

Find the least common multiple (LCM) of 18 and 24.
If you missed this problem, review Example 1.3.

Example 1.3 in the book ↗

Answer

72

Be Prepared 6.3 §6.1 source ↗

Multiply:

−3 a (7 a + 8 b) .

If you missed this problem, review Example 5.26.

Example 5.26 in the book ↗

Answer

−21 a^{2} - 24 a b

Try It 6.1 §6.1 source ↗

Find the greatest common factor:

25 m^{4},

35 m^{3},

20 m^{2} .

Answer

5 m^{2}

Try It 6.2 §6.1 source ↗

Find the greatest common factor:

14 x^{3},

70 x^{2},

105 x .

Answer

7 x

Try It 6.3 §6.1 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 6.4 §6.1 source ↗

Factor:

3 p^{3} -

6 p^{2} q +

9 p q^{3} .

Answer

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

Try It 6.5 §6.1 source ↗

Factor:

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

Answer

2 x^{2} (x + 6)

Try It 6.6 §6.1 source ↗

Factor:

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

Answer

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

Try It 6.7 §6.1 source ↗

Factor:

15 x^{3} y -

3 x^{2} y^{2} +

6 x y^{3} .

Answer

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

Try It 6.8 §6.1 source ↗

Factor:

8 a^{3} b +

2 a^{2} b^{2} -

6 a b^{3} .

Answer

2 a b (4 a^{2} + a b - 3 b^{2})

Try It 6.9 §6.1 source ↗

Factor:

−4 b^{3} +

16 b^{2} -

8 b .

Answer

−4 b (b^{2} - 4 b + 2)

Try It 6.10 §6.1 source ↗

Factor:

−7 a^{3} +

21 a^{2} -

14 a .

Answer

−7 a (a^{2} - 3 a + 2)

Try It 6.11 §6.1 source ↗

Factor:

4 m (m + 3) -

7 (m + 3) .

Answer

(m + 3) (4 m - 7)

Try It 6.12 §6.1 source ↗

Factor:

8 n (n - 4) +

5 (n - 4) .

Answer

(n - 4) (8 n + 5)

Try It 6.13 §6.1 source ↗

Factor by grouping:

x y +

8 y +

3 x +

24 .

Answer

(x + 8) (y + 3)

Try It 6.14 §6.1 source ↗

Factor by grouping:

a b +

7 b +

8 a +

56 .

Answer

(a + 7) (b + 8)

Try It 6.15 §6.1 source ↗

Factor by grouping: ⓐ

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

ⓑ

20 x^{2} -

16 x -

15 x +

12 .

Answer

ⓐ

(x - 5) (x + 2)

ⓑ

(5 x - 4) (4 x - 3)

Try It 6.16 §6.1 source ↗

Factor by grouping: ⓐ

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

ⓑ

42 m^{2} -

18 m -

35 m +

15 .

Answer

ⓐ

(y + 4) (y - 7)

ⓑ

(7 m - 3) (6 m - 5)

6.1-1 §6.1 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

6.1-3 §6.1 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}

6.1-5 §6.1 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

6.1-7 §6.1 source ↗

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$35 x^{3} y^{2},$ $10 x^{4} y,$ $5 x^{5} y^{3}$

Answer

5 x^{3} y

6.1-9 §6.1 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)

6.1-11 §6.1 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)

6.1-13 §6.1 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)

6.1-15 §6.1 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)

6.1-17 §6.1 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)

6.1-19 §6.1 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)

6.1-21 §6.1 source ↗

Factor the Greatest Common Factor from a Polynomial

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

$24 x^{3} -$ $12 x^{2} +$ $15 x$

Answer

3 x (8 x^{2} - 4 x + 5)

6.1-23 §6.1 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)

6.1-25 §6.1 source ↗

Factor the Greatest Common Factor from a Polynomial

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

$20 x^{3} y -$ $4 x^{2} y^{2} +$ $12 x y^{3}$

Answer

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

6.1-27 §6.1 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)

6.1-29 §6.1 source ↗

Factor the Greatest Common Factor from a Polynomial

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

$−2 x^{3} +$ $18 x^{2} -$ $8 x$

Answer

−2 x (x^{2} - 9 x + 4)

6.1-31 §6.1 source ↗

Factor the Greatest Common Factor from a Polynomial

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

$−4 p^{3} q -$ $12 p^{2} q^{2} +$ $16 p q^{2}$

Answer

−4 p q (p^{2} + 3 p q - 4 q)

6.1-33 §6.1 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)

6.1-35 §6.1 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)

6.1-37 §6.1 source ↗

Factor by Grouping

In the following exercises, factor by grouping.

$a b + 5 a + 3 b + 15$

Answer

(b + 5) (a + 3)

6.1-39 §6.1 source ↗

Factor by Grouping

In the following exercises, factor by grouping.

$8 y^{2} + y + 40 y + 5$

Answer

(y + 5) (8 y + 1)

6.1-41 §6.1 source ↗

Factor by Grouping

In the following exercises, factor by grouping.

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

Answer

(u + 2) (v - 9)

6.1-43 §6.1 source ↗

Factor by Grouping

In the following exercises, factor by grouping.

$u^{2} - u + 6 u - 6$

Answer

(u - 1) (u + 6)

6.1-45 §6.1 source ↗

Factor by Grouping

In the following exercises, factor by grouping.

$9 p^{2} +$ $12 p -$ $15 p -$ $20$

Answer

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

6.1-47 §6.1 source ↗

Factor by Grouping

In the following exercises, factor by grouping.

$m n - 6 m - 4 n + 24$

Answer

(n - 6) (m - 4)

6.1-49 §6.1 source ↗

Factor by Grouping

In the following exercises, factor by grouping.

$2 x^{2} -$ $14 x -$ $5 x +$ $35$

Answer

(x - 7) (2 x - 5)

6.1-51 §6.1 source ↗

Mixed Practice

In the following exercises, factor.

$−18 x y^{2} -$ $27 x^{2} y$

Answer

−9 x y (2 y + 3 x)

6.1-53 §6.1 source ↗

Mixed Practice

In the following exercises, factor.

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

Answer

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

6.1-55 §6.1 source ↗

Mixed Practice

In the following exercises, factor.

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

Answer

(x + y) (x + 5)

6.1-57 §6.1 source ↗

Mixed Practice

In the following exercises, factor.

What does it mean to say a polynomial is in factored form?

Answer

Answers will vary.

6.1-59 §6.1 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 6.4 §6.2 source ↗

Find all the factors of 72.
If you missed this problem, review Example 1.2.

Example 1.2 in the book ↗

Answer

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Be Prepared 6.5 §6.2 source ↗

Find the product:

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

If you missed this problem, review Example 5.28.

Example 5.28 in the book ↗

Answer

6 y^{2} + 23 y + 20

Be Prepared 6.6 §6.2 source ↗

Simplify:

−9 (6);

−9 (−6) .

If you missed this problem, review Example 1.18.

Example 1.18 in the book ↗

Answer

−54, 54

Try It 6.17 §6.2 source ↗

Factor:

q^{2} + 10 q + 24 .

Answer

(q + 4) (q + 6)

Try It 6.18 §6.2 source ↗

Factor:

t^{2} + 14 t + 24 .

Answer

(t + 2) (t + 12)

Try It 6.19 §6.2 source ↗

Factor:

u^{2} - 9 u + 18 .

Answer

(u - 3) (u - 6)

Try It 6.20 §6.2 source ↗

Factor:

y^{2} - 16 y + 63 .

Answer

(y - 7) (y - 9)

Try It 6.21 §6.2 source ↗

Factor:

9 m + m^{2} + 18 .

Answer

(m + 3) (m + 6)

Try It 6.22 §6.2 source ↗

Factor:

−7 n + 12 + n^{2} .

Answer

(n - 3) (n - 4)

Try It 6.23 §6.2 source ↗

Factor:

a^{2} -

11 a b +

10 b^{2} .

Answer

(a - b) (a - 10 b)

Try It 6.24 §6.2 source ↗

Factor:

m^{2} -

13 m n +

12 n^{2} .

Answer

(m - n) (m - 12 n)

Try It 6.25 §6.2 source ↗

Factor:

x^{2} -

7 x y -

10 y^{2} .

Answer

prime

Try It 6.26 §6.2 source ↗

Factor:

p^{2} +

15 p q +

20 q^{2} .

Answer

prime

Try It 6.27 §6.2 source ↗

Factor completely:

5 x^{3} +

15 x^{2} -

20 x .

Answer

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

Try It 6.28 §6.2 source ↗

Factor completely:

6 y^{3} +

18 y^{2} -

60 y .

Answer

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

Try It 6.29 §6.2 source ↗

Factor completely using trial and error:

2 a^{2} + 5 a + 3 .

Answer

(a + 1) (2 a + 3)

Try It 6.30 §6.2 source ↗

Factor completely using trial and error:

4 b^{2} + 5 b + 1 .

Answer

(b + 1) (4 b + 1)

Try It 6.31 §6.2 source ↗

Factor completely using trial and error:

8 x^{2} - 14 x + 3 .

Answer

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

Try It 6.32 §6.2 source ↗

Factor completely using trial and error:

10 y^{2} - 37 y + 7 .

Answer

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

Try It 6.33 §6.2 source ↗

Factor completely using trial and error

18 x^{2} -

3 x y -

10 y^{2} .

Answer

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

Try It 6.34 §6.2 source ↗

Factor completely using trial and error:

30 x^{2} -

53 x y -

21 y^{2} .

Answer

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

Try It 6.35 §6.2 source ↗

Factor completely using trial and error:

15 n^{3} -

85 n^{2} +

100 n .

Answer

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

Try It 6.36 §6.2 source ↗

Factor completely using trial and error:

56 q^{3} +

320 q^{2} -

96 q .

Answer

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

Try It 6.37 §6.2 source ↗

Factor using the ‘ac’ method:

6 x^{2} + 13 x + 2 .

Answer

(x + 2) (6 x + 1)

Try It 6.38 §6.2 source ↗

Factor using the ‘ac’ method:

4 y^{2} + 8 y + 3 .

Answer

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

Try It 6.39 §6.2 source ↗

Factor using the ‘ac’ method:

16 x^{2} -

32 x +

12 .

Answer

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

Try It 6.40 §6.2 source ↗

Factor using the ‘ac’ method:

18 w^{2} -

39 w +

18 .

Answer

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

Try It 6.41 §6.2 source ↗

Factor by substitution:

h^{4} + 4 h^{2} - 12 .

Answer

(h^{2} - 2) (h^{2} + 6)

Try It 6.42 §6.2 source ↗

Factor by substitution:

y^{4} - y^{2} - 20 .

Answer

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

Try It 6.43 §6.2 source ↗

Factor by substitution:

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

6 (x - 5) +

8 .

Answer

(x - 3) (x - 1)

Try It 6.44 §6.2 source ↗

Factor by substitution:

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

8 (y - 4) +

15 .

Answer

(y - 1) (y + 1)

6.2-61 §6.2 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)

6.2-63 §6.2 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)

6.2-65 §6.2 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)

6.2-67 §6.2 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)

6.2-69 §6.2 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)

6.2-71 §6.2 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)

6.2-73 §6.2 source ↗

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

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

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

Answer

(p - 1) (p + 6)

6.2-75 §6.2 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)

6.2-77 §6.2 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)

6.2-79 §6.2 source ↗

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)

6.2-81 §6.2 source ↗

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)

6.2-83 §6.2 source ↗

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)

6.2-85 §6.2 source ↗

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

6.2-87 §6.2 source ↗

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

6.2-89 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error

In the following exercises, factor completely using trial and error.

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

Answer

p (p - 10) (p + 2)

6.2-91 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error

In the following exercises, factor completely using trial and error.

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

Answer

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

6.2-93 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error

In the following exercises, factor completely using trial and error.

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

Answer

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

6.2-95 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error

In the following exercises, factor completely using trial and error.

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

Answer

(2 t + 5) (t + 1)

6.2-97 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error

In the following exercises, factor completely using trial and error.

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

Answer

(11 x + 1) (x + 3)

6.2-99 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error

In the following exercises, factor completely using trial and error.

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

Answer

(4 w - 1) (w - 1)

6.2-101 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error

In the following exercises, factor completely using trial and error.

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

Answer

(4 q + 1) (q - 2)

6.2-103 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error

In the following exercises, factor completely using trial and error.

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

Answer

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

6.2-105 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error

In the following exercises, factor completely using trial and error.

$4 a^{2} +$ $17 a b -$ $15 b^{2}$

Answer

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

6.2-107 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error

In the following exercises, factor completely using trial and error.

$−16 x^{2} -$ $32 x -$ $16$

Answer

−16 (x + 1) (x + 1)

6.2-109 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error

In the following exercises, factor completely using trial and error.

$−30 q^{3} -$ $140 q^{2} -$ $80 q$

Answer

−10 q (3 q + 2) (q + 4)

6.2-111 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ using the ‘ac’ Method

In the following exercises, factor using the ‘ac’ method.

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

Answer

(5 n + 1) (n + 4)

6.2-113 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ using the ‘ac’ Method

In the following exercises, factor using the ‘ac’ method.

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

Answer

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

6.2-115 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ using the ‘ac’ Method

In the following exercises, factor using the ‘ac’ method.

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

Answer

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

6.2-117 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ using the ‘ac’ Method

In the following exercises, factor using the ‘ac’ method.

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

Answer

(2 n + 3) (n - 15)

6.2-119 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ using the ‘ac’ Method

In the following exercises, factor using the ‘ac’ method.

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

Answer

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

6.2-121 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ using the ‘ac’ Method

In the following exercises, factor using the ‘ac’ method.

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

Answer

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

6.2-123 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ using the ‘ac’ Method

In the following exercises, factor using the ‘ac’ method.

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

Answer

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

6.2-125 §6.2 source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ using the ‘ac’ Method

In the following exercises, factor using the ‘ac’ method.

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

Answer

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

6.2-127 §6.2 source ↗

Factor Using Substitution

In the following exercises, factor using substitution.

$x^{4} - 6 x^{2} - 7$

Answer

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

6.2-129 §6.2 source ↗

Factor Using Substitution

In the following exercises, factor using substitution.

$x^{4} - 3 x^{2} - 28$

Answer

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

6.2-131 §6.2 source ↗

Factor Using Substitution

In the following exercises, factor using substitution.

${(x - 3)}^{2} -$ $5 (x - 3) -$ $36$

Answer

(x - 12) (x + 1)

6.2-133 §6.2 source ↗

Factor Using Substitution

In the following exercises, factor using substitution.

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

Answer

(3 y - 4) (3 y - 1)

6.2-135 §6.2 source ↗

Mixed Practice

In the following exercises, factor each expression using any method.

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

Answer

(u - 6) (u - 6)

6.2-137 §6.2 source ↗

Mixed Practice

In the following exercises, factor each expression using any method.

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

Answer

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

6.2-139 §6.2 source ↗

Mixed Practice

In the following exercises, factor each expression using any method.

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

Answer

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

6.2-141 §6.2 source ↗

Mixed Practice

In the following exercises, factor each expression using any method.

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

Answer

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

6.2-143 §6.2 source ↗

Mixed Practice

In the following exercises, factor each expression using any method.

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

Answer

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

6.2-145 §6.2 source ↗

Mixed Practice

In the following exercises, factor each expression using any method.

$3 p^{2} + 21 p$

Answer

3 p (p + 7)

6.2-147 §6.2 source ↗

Mixed Practice

In the following exercises, factor each expression using any method.

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

Answer

6 (r + 2) (r + 3)

6.2-149 §6.2 source ↗

Mixed Practice

In the following exercises, factor each expression using any method.

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

Answer

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

6.2-151 §6.2 source ↗

Mixed Practice

In the following exercises, factor each expression using any method.

$x^{4} - 4 x^{2} - 12$

Answer

(x^{2} + 2) (x^{2} - 6)

6.2-153 §6.2 source ↗

Mixed Practice

In the following exercises, factor each expression using any method.

${(x + 3)}^{2} -$ $9 (x + 3) -$ $36$

Answer

(x - 9) (x + 6)

6.2-155 §6.2 source ↗

Mixed Practice

In the following exercises, factor each expression using any method.

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 will vary.

6.2-157 §6.2 source ↗

Mixed Practice

In the following exercises, factor each expression using any method.

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 will vary.

Be Prepared 6.7 §6.3 source ↗

Simplify:

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

If you missed this problem, review Example 5.18.

Example 5.18 in the book ↗

Answer

27 x^{6}

Be Prepared 6.8 §6.3 source ↗

Multiply:

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

If you missed this problem, review Example 5.32.

Example 5.32 in the book ↗

Answer

m^{2} + 8 m + 16

Be Prepared 6.9 §6.3 source ↗

Multiply:

(x - 3) (x + 3) .

If you missed this problem, review Example 5.33.

Example 5.33 in the book ↗

Answer

x^{2} - 9

Try It 6.45 §6.3 source ↗

Factor:

4 x^{2} + 12 x + 9 .

Answer

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

Try It 6.46 §6.3 source ↗

Factor:

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

Answer

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

Try It 6.47 §6.3 source ↗

Factor:

64 y^{2} -

80 y +

25 .

Answer

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

Try It 6.48 §6.3 source ↗

Factor:

16 z^{2} -

72 z +

81 .

Answer

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

Try It 6.49 §6.3 source ↗

Factor:

49 x^{2} +

84 x y +

36 y^{2} .

Answer

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

Try It 6.50 §6.3 source ↗

Factor:

64 m^{2} +

112 m n +

49 n^{2} .

Answer

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

Try It 6.51 §6.3 source ↗

Factor:

8 x^{2} y -

24 x y +

18 y .

Answer

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

Try It 6.52 §6.3 source ↗

Factor:

27 p^{2} q +

90 p q +

75 q .

Answer

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

Try It 6.53 §6.3 source ↗

Factor:

121 m^{2} - 1 .

Answer

(11 m - 1) (11 m + 1)

Try It 6.54 §6.3 source ↗

Factor:

81 y^{2} - 1 .

Answer

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

Try It 6.55 §6.3 source ↗

Factor:

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

Answer

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

Try It 6.56 §6.3 source ↗

Factor:

121 p^{2} - 9 q^{2} .

Answer

(11 p - 3 q) (11 p + 3 q)

Try It 6.57 §6.3 source ↗

Factor:

2 x^{4} y^{2} - 32 y^{2} .

Answer

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

Try It 6.58 §6.3 source ↗

Factor:

7 a^{4} c^{2} -

7 b^{4} c^{2} .

Answer

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

Try It 6.59 §6.3 source ↗

Factor:

x^{2} -

10 x +

25 -

y^{2} .

Answer

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

Try It 6.60 §6.3 source ↗

Factor:

x^{2} +

6 x +

9 -

4 y^{2} .

Answer

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

Try It 6.61 §6.3 source ↗

Factor:

x^{3} + 27 .

Answer

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

Try It 6.62 §6.3 source ↗

Factor:

y^{3} + 8 .

Answer

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

Try It 6.63 §6.3 source ↗

Factor:

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

Answer

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

Try It 6.64 §6.3 source ↗

Factor:

1000 m^{3} -

125 n^{3} .

Answer

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

Try It 6.65 §6.3 source ↗

Factor:

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

Answer

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

Try It 6.66 §6.3 source ↗

Factor:

432 c^{3} +

686 d^{3} .

Answer

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

Try It 6.67 §6.3 source ↗

Factor:

{(y + 1)}^{3} -

27 y^{3} .

Answer

(−2 y + 1) (13 y^{2} + 5 y + 1)

Try It 6.68 §6.3 source ↗

Factor:

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

125 n^{3} .

Answer

(−4 n + 3) (31 n^{2} + 21 n + 9)

6.3-159 §6.3 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

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

Answer

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

6.3-161 §6.3 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

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

Answer

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

6.3-163 §6.3 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

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

Answer

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

6.3-165 §6.3 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

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

Answer

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

6.3-167 §6.3 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

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

Answer

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

6.3-169 §6.3 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

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

Answer

(10 y - 1)^{2}

6.3-171 §6.3 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

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

Answer

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

6.3-173 §6.3 source ↗

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

$75 u^{4} -$ $30 u^{3} v +$ $3 u^{2} v^{2}$

Answer

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

6.3-175 §6.3 source ↗

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

$25 v^{2} - 1$

Answer

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

6.3-177 §6.3 source ↗

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

$4 - 49 x^{2}$

Answer

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

6.3-179 §6.3 source ↗

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

$6 p^{2} q^{2} - 54 p^{2}$

Answer

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

6.3-181 §6.3 source ↗

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

$24 p^{2} + 54$

Answer

6 (4 p^{2} + 9)

6.3-183 §6.3 source ↗

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

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

Answer

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

6.3-185 §6.3 source ↗

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

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

Answer

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

6.3-187 §6.3 source ↗

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

$16 z^{4} - 1$

Answer

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

6.3-189 §6.3 source ↗

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

$162 a^{4} b^{2} -$ $32 b^{2}$

Answer

2 b^{2} (3 a - 2) (3 a + 2) (9 a^{2} + 4)

6.3-191 §6.3 source ↗

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

$x^{2} -$ $16 x +$ $64 -$ $y^{2}$

Answer

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

6.3-193 §6.3 source ↗

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

$a^{2} + 6 a + 9 - 9 b^{2}$

Answer

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

6.3-195 §6.3 source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

$x^{3} + 125$

Answer

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

6.3-197 §6.3 source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

$z^{6} - 27$

Answer

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

6.3-199 §6.3 source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

$8 - 343 t^{3}$

Answer

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

6.3-201 §6.3 source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

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

Answer

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

6.3-203 §6.3 source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

$216 a^{3} + 125 b^{3}$

Answer

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

6.3-205 §6.3 source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

$7 k^{3} + 56$

Answer

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

6.3-207 §6.3 source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

$2 x^{2} - 16 x^{2} y^{3}$

Answer

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

6.3-209 §6.3 source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

${(x + 3)}^{3} + 8 x^{3}$

Answer

9 (x + 1) (x^{2} + 3)

6.3-211 §6.3 source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

${(y - 5)}^{3} - 64 y^{3}$

Answer

− (3 y + 5) (21 y^{2} - 30 y + 25)

6.3-213 §6.3 source ↗

Mixed Practice

In the following exercises, factor completely.

$64 a^{2} - 25$

Answer

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

6.3-215 §6.3 source ↗

Mixed Practice

In the following exercises, factor completely.

$27 q^{2} - 3$

Answer

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

6.3-217 §6.3 source ↗

Mixed Practice

In the following exercises, factor completely.

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

Answer

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

6.3-219 §6.3 source ↗

Mixed Practice

In the following exercises, factor completely.

$8 p^{2} + 2$

Answer

2 (4 p^{2} + 1)

6.3-221 §6.3 source ↗

Mixed Practice

In the following exercises, factor completely.

$125 - 8 y^{3}$

Answer

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

6.3-223 §6.3 source ↗

Mixed Practice

In the following exercises, factor completely.

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

Answer

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

6.3-225 §6.3 source ↗

Mixed Practice

In the following exercises, factor completely.

$x^{2} -$ $10 x +$ $25 -$ $y^{2}$

Answer

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

6.3-227 §6.3 source ↗

Mixed Practice

In the following exercises, factor completely.

${(x + 1)}^{3} + 8 x^{3}$

Answer

(3 x + 1) (3 x^{2} + 1)

6.3-229 §6.3 source ↗

Mixed Practice

In the following exercises, factor completely.

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

Answer

Answers will vary.

6.3-231 §6.3 source ↗

Mixed Practice

In the following exercises, factor completely.

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

Answer

Answers will vary.

Try It 6.69 §6.4 source ↗

Factor completely:

8 y^{3} +

16 y^{2} -

24 y .

Answer

8 y (y - 1) (y + 3)

Try It 6.70 §6.4 source ↗

Factor completely:

5 y^{3} -

15 y^{2} -

270 y .

Answer

5 y (y - 9) (y + 6)

Try It 6.71 §6.4 source ↗

Factor completely:

16 x^{3} - 36 x .

Answer

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

Try It 6.72 §6.4 source ↗

Factor completely:

27 y^{2} - 48 .

Answer

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

Try It 6.73 §6.4 source ↗

Factor completely:

4 x^{2} +

20 x y +

25 y^{2} .

Answer

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

Try It 6.74 §6.4 source ↗

Factor completely:

9 x^{2} -

24 x y +

16 y^{2} .

Answer

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

Try It 6.75 §6.4 source ↗

Factor completely:

50 x^{3} y + 72 x y .

Answer

2 x y (25 x^{2} + 36)

Try It 6.76 §6.4 source ↗

Factor completely:

27 x y^{3} + 48 x y .

Answer

3 x y (9 y^{2} + 16)

Try It 6.77 §6.4 source ↗

Factor completely:

250 m^{3} +

432 n^{3} .

Answer

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

Try It 6.78 §6.4 source ↗

Factor completely:

2 p^{3} + 54 q^{3} .

Answer

2 (p + 3 q) (p^{2} - 3 p q + 9 q^{2})

Try It 6.79 §6.4 source ↗

Factor completely:

4 a^{5} b - 64 a b .

Answer

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

Try It 6.80 §6.4 source ↗

Factor completely:

7 x y^{5} - 7 x y .

Answer

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

Try It 6.81 §6.4 source ↗

Factor completely:

6 x^{2} -

12 x c +

6 b x -

12 b c .

Answer

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

Try It 6.82 §6.4 source ↗

Factor completely:

16 x^{2} +

24 x y -

4 x -

6 y .

Answer

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

Try It 6.83 §6.4 source ↗

Factor completely:

4 p^{2} q -

16 p q +

12 q .

Answer

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

Try It 6.84 §6.4 source ↗

Factor completely:

6 p q^{2} -

9 p q -

6 p .

Answer

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

Try It 6.85 §6.4 source ↗

Factor completely:

4 x^{2} -

12 x y +

9 y^{2} -

25 .

Answer

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

Try It 6.86 §6.4 source ↗

Factor completely:

16 x^{2} -

24 x y +

9 y^{2} -

64 .

Answer

(4 x - 3 y - 8) (4 x - 3 y + 8)

6.4-233 §6.4 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)

6.4-235 §6.4 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)

6.4-237 §6.4 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)

6.4-239 §6.4 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)

6.4-241 §6.4 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}

6.4-243 §6.4 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}

6.4-245 §6.4 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)

6.4-247 §6.4 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$3 x^{4} y - 81 x y$

Answer

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

6.4-249 §6.4 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)

6.4-251 §6.4 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$5 x^{5} y^{2} - 80 x y^{2}$

Answer

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

6.4-253 §6.4 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)

6.4-255 §6.4 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)

6.4-257 §6.4 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$4 u^{5} + 4 u^{2} v^{3}$

Answer

4 u^{2} (u + v) (u^{2} - u v + v^{2})

6.4-259 §6.4 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

6.4-261 §6.4 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)

6.4-263 §6.4 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$36 x^{2} y +$ $15 x y -$ $6 y$

Answer

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

6.4-265 §6.4 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

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

Answer

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

6.4-267 §6.4 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$y^{6} - 1$

Answer

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

6.4-269 §6.4 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$9 x^{2} -$ $6 x y +$ $y^{2} -$ $49$

Answer

(3 x - y + 7) (3 x - y - 7)

6.4-271 §6.4 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

${(3 x + 1)}^{2} -$ $6 (3 x + 1) +$ $9$

Answer

(3 x - 2)^{2}

6.4-273 §6.4 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

Explain what it mean to factor a polynomial completely.

Answer

Answers will vary.

6.4-275 §6.4 source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

Of all the factoring methods covered in this chapter (GCF, grouping, undo FOIL, ‘ac’ method, special products) which is the easiest for you? Which is the hardest? Explain your answers.

Answer

Answers will vary.

Be Prepared 6.10 §6.5 source ↗

Solve:

5 y - 3 = 0 .

If you missed this problem, review Example 2.2.

Example 2.2 in the book ↗

Answer

y = \frac{3}{5}

Be Prepared 6.11 §6.5 source ↗

Factor completely:

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

If you missed this problem, review Example 3.48.

Example 3.48 in the book ↗

Answer

n (n - 11) (n + 2)

Be Prepared 6.12 §6.5 source ↗

If

f (x) = 8 x - 16,

find

f (3)

and solve

f (x) = 0 .

If you missed this problem, review Example 3.59.

Example 3.59 in the book ↗

Answer

8; x = 2

Try It 6.87 §6.5 source ↗

Solve:

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

0 .

Answer

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

Try It 6.88 §6.5 source ↗

Solve:

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

0 .

Answer

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

Try It 6.89 §6.5 source ↗

Solve:

3 c^{2} = 10 c - 8 .

Answer

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

Try It 6.90 §6.5 source ↗

Solve:

2 d^{2} - 5 d = 3 .

Answer

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

Try It 6.91 §6.5 source ↗

Solve:

25 p^{2} = 49 .

Answer

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

Try It 6.92 §6.5 source ↗

Solve:

36 x^{2} = 121 .

Answer

x =

\frac{11}{6},

x =

-

\frac{11}{6}

Try It 6.93 §6.5 source ↗

Solve:

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

12 m .

Answer

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

Try It 6.94 §6.5 source ↗

Solve:

(k + 1) (k - 1) =

8 .

Answer

k = 3, k = −3

Try It 6.95 §6.5 source ↗

Solve:

18 a^{2} -

30 =

−33 a .

Answer

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

Try It 6.96 §6.5 source ↗

Solve:

123 b =

−6 -

60 b^{2} .

Answer

b = −2, b = - \frac{1}{20}

Try It 6.97 §6.5 source ↗

Solve:

8 x^{3} =

24 x^{2} -

18 x .

Answer

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

Try It 6.98 §6.5 source ↗

Solve:

16 y^{2} =

32 y^{3} +

2 y .

Answer

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

Try It 6.99 §6.5 source ↗

For the function $f (x) =$ $x^{2} -$ $2 x -$ $8,$

ⓐ find x when $f (x) = 7$ ⓑ Find two points that lie on the graph of the function.

Answer

ⓐ

x = −3

or

x = 5

ⓑ

(−3, 7)

(5, 7)

Try It 6.100 §6.5 source ↗

For the function $f (x) =$ $x^{2} -$ $8 x +$ $3,$

ⓐ find x when $f (x) = −4$ ⓑ Find two points that lie on the graph of the function.

Answer

ⓐ

x = 1

or

x = 7

ⓑ

(1, −4)

(7, −4)

Try It 6.101 §6.5 source ↗

For the function $f (x) =$ $2 x^{2} -$ $7 x +$ $5,$ find

ⓐ the zeros of the function, ⓑ any x-intercepts of the graph of the function, ⓒ any y-intercepts of the graph of the function.

Answer

ⓐ

x = 1

or

x = \frac{5}{2}

ⓑ

(1, 0),

(\frac{5}{2}, 0)

ⓒ

(0, 5)

Try It 6.102 §6.5 source ↗

For the function $f (x) =$ $6 x^{2} +$ $13 x -$ $15,$ find

ⓐ the zeros of the function, ⓑ any x-intercepts of the graph of the function, ⓒ any y-intercepts of the graph of the function.

Answer

ⓐ

x = −3

or

x = \frac{5}{6}

ⓑ

(−3, 0),

(\frac{5}{6}, 0)

ⓒ

(0, −15)

Try It 6.103 §6.5 source ↗

The product of two consecutive odd integers is 255. Find the integers.

Answer

−15, −17

and 15, 17

Try It 6.104 §6.5 source ↗

The product of two consecutive odd integers is 483 Find the integers.

Answer

−23, −21

and 21, 23

Try It 6.105 §6.5 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

The width is 5 feet and length is 6 feet.

Try It 6.106 §6.5 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

The width of the patio is 12 feet and the length is 15 feet.

Try It 6.107 §6.5 source ↗

Justine wants to put a deck in the corner of her backyard in the shape of a right triangle. The length of one side of the deck is 7 feet more than the other side. The hypotenuse is 13. Find the lengths of the two sides of the deck.

Answer

5 feet and 12 feet

Try It 6.108 §6.5 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 the other leg. Find the lengths of the hypotenuse and the other leg.

Answer

The other leg is 24 feet and the hypotenuse is 25 feet.

Try It 6.109 §6.5 source ↗

Genevieve is going to throw a rock from the top a trail overlooking the ocean. When she throws the rock upward from 160 feet above the ocean, the function $h (t) =$ $−16 t^{2} +$ $48 t +$ $160$ models the height, h, of the rock above the ocean as a function of time, t. Find:

ⓐ the zeros of this function which tell us when the rock will hit the ocean, ⓑ when the rock will be 160 feet above the ocean, ⓒ the height of the rock at $t = 1.5$ seconds.

Answer

Try It 6.110 §6.5 source ↗

Calib is going to throw his lucky penny from his balcony on a cruise ship. When he throws the penny upward from 128 feet above the ground, the function $h (t) =$ $−16 t^{2} +$ $32 t +$ $128$ models the height, h, of the penny above the ocean as a function of time, t. Find:

ⓐ the zeros of this function which is when the penny will hit the ocean, ⓑ when the penny will be 128 feet above the ocean, ⓒ the height the penny will be at $t = 1$ seconds which is when the penny will be at its highest point.

Answer

6.5-277 §6.5 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

6.5-279 §6.5 source ↗

Use the Zero Product Property

In the following exercises, solve.

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

Answer

m = 0, m = 5 / 12

6.5-281 §6.5 source ↗

Use the Zero Product Property

In the following exercises, solve.

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

Answer

x = 1 / 2

6.5-283 §6.5 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

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

Answer

a = \frac{−4}{5}, a = 6

6.5-285 §6.5 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

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

Answer

m = 5 / 4, m = 3

6.5-287 §6.5 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

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

Answer

a = −1, a = 0

6.5-289 §6.5 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$49 m^{2} = 144$

Answer

m =

12 / 7,

m =

−12 / 7

6.5-291 §6.5 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$16 y^{2} = 81$

Answer

y =

−9 / 4,

y =

9 / 4

6.5-293 §6.5 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$121 n^{2} = 36$

Answer

n =

−6 / 11,

n =

6 / 11

6.5-295 §6.5 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$(x + 6) (x - 3) =$ $−8$

Answer

x = 2, x = −5

6.5-297 §6.5 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$(2 x + 1) (x - 3) =$ $−4 x$

Answer

x = 3 / 2, x = −1

6.5-299 §6.5 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$(3 x - 2) (x + 4) =$ $12 x$

Answer

x = 2, x = −4 / 3

6.5-301 §6.5 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$20 x^{2} -$ $60 x =$ $−45$

Answer

x = 3 / 2

6.5-303 §6.5 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$15 x^{2} - 10 x = 40$

Answer

x = 2, x = −4 / 3

6.5-305 §6.5 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$18 x^{2} - 9 = −21 x$

Answer

x =

−3 / 2,

x =

1 / 3

6.5-307 §6.5 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

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

Answer

p = 0, p = ¾

6.5-309 §6.5 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$2 x^{3} +$ $72 x =$ $24 x^{2}$

Answer

x = 0, x = 6

6.5-311 §6.5 source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$36 x^{3} +$ $24 x^{2} =$ $−4 x$

Answer

x = 0, x = –1 / 3

6.5-313 §6.5 source ↗

Solve Equations with Polynomial Functions

In the following exercises, solve.

For the function, $f (x) =$ $x^{2} -$ $8 x +$ $8,$ ⓐ find when $f (x) = −4$ ⓑ Use this information to find two points that lie on the graph of the function.

Answer

ⓐ

x = 2

or

x = 6

ⓑ

(2, −4)

(6, −4)

6.5-315 §6.5 source ↗

Solve Equations with Polynomial Functions

In the following exercises, solve.

For the function, $f (x) =$ $8 x^{2} -$ $18 x +$ $5,$ ⓐ find when $f (x) = −4$ ⓑ Use this information to find two points that lie on the graph of the function.

Answer

ⓐ

x = \frac{3}{2}

or

x = \frac{3}{4}

ⓑ

(\frac{3}{2}, −4)

(\frac{3}{4}, −4)

6.5-317 §6.5 source ↗

In the following exercises, for each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.

$f (x) = 9 x^{2} - 4$

Answer

ⓐ

x = \frac{2}{3}

or

x = - \frac{2}{3}

ⓑ

(\frac{2}{3}, 0)

,

(- \frac{2}{3}, 0)

ⓒ

(0, −4)

6.5-319 §6.5 source ↗

In the following exercises, for each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.

$f (x) =$ $6 x^{2} -$ $7 x -$ $5$

Answer

ⓐ

x = \frac{5}{3}

or

x = - \frac{1}{2}

ⓑ

(\frac{5}{3}, 0)

,

(- \frac{1}{2}, 0)

ⓒ

(0, −5)

6.5-321 §6.5 source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

The product of two consecutive odd integers is 143. Find the integers.

Answer

−13, −11

and 11, 13

6.5-323 §6.5 source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

The product of two consecutive even integers is 168. Find the integers.

Answer

−14, −12

and 12, 14

6.5-325 §6.5 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

Width: 4 feet; Length: 7 feet.

6.5-327 §6.5 source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

The area of a bulletin board is 55 square feet. The length is four feet less than three times the width. Find the length and the width of the a bulletin board.

Answer

Width: 5 feet; Length: 11 feet.

6.5-329 §6.5 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

The sides are 6 feet and 8 feet.

6.5-331 §6.5 source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

A reflecting pool is shaped like a right triangle, with one leg along the wall of a building. The hypotenuse is 9 feet longer than the side along the building. The third side is 7 feet longer than the side along the building. Find the lengths of all three sides of the reflecting pool.

Answer

The building side is 8 feet, the hypotenuse is 17 feet, and the third side is 15 feet.

6.5-333 §6.5 source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

Juli is going to launch a model rocket in her back yard. When she launches the rocket, the function $h (t) =$ $−16 t^{2} +$ $32 t$ models the height, h, of the rocket above the ground as a function of time, t. Find:

ⓐ the zeros of this function, which tell us when the rocket will be on the ground. ⓑ the time the rocket will be 16 feet above the ground.

Answer

ⓐ 0 seconds and 2 seconds ⓑ 1 second

6.5-335 §6.5 source ↗

Solve Applications Modeled by Quadratic Equations

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 will vary.

6.R-337 §6.R source ↗

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$12 a^{2} b^{3},$ $15 a b^{2}$

Answer

3 a b^{2}

6.R-339 §6.R source ↗

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$15 y^{3},$ $21 y^{2},$ $30 y$

Answer

3 y

6.R-341 §6.R source ↗

Factor the Greatest Common Factor from a Polynomial

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

$35 y + 84$

Answer

7 (5 y + 12)

6.R-343 §6.R source ↗

Factor the Greatest Common Factor from a Polynomial

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

$18 x^{3} - 15 x$

Answer

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

6.R-345 §6.R source ↗

Factor the Greatest Common Factor from a Polynomial

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

$4 x^{3} -$ $12 x^{2} +$ $16 x$

Answer

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

6.R-347 §6.R source ↗

Factor the Greatest Common Factor from a Polynomial

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

$−3 x^{3} +$ $27 x^{2} -$ $12 x$

Answer

−3 x (x^{2} - 9 x + 4)

6.R-349 §6.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)

6.R-351 §6.R source ↗

Factor by Grouping

In the following exercises, factor by grouping.

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

Answer

(x - 3) (x + 7)

6.R-353 §6.R source ↗

Factor by Grouping

In the following exercises, factor by grouping.

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

Answer

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

6.R-355 §6.R source ↗

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

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

$a^{2} + 14 a + 33$

Answer

(a + 3) (a + 11)

6.R-357 §6.R source ↗

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

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

$m^{2} + 3 m - 54$

Answer

(m + 9) (m - 6)

6.R-359 §6.R source ↗

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)

6.R-361 §6.R source ↗

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)

6.R-363 §6.R source ↗

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

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

Answer

Prime

6.R-365 §6.R source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error

In the following exercises, factor completely using trial and error.

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

Answer

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

6.R-367 §6.R source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error

In the following exercises, factor completely using trial and error.

$5 y^{2} + 14 y + 9$

Answer

(5 y + 9) (y + 1)

6.R-369 §6.R source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error

In the following exercises, factor completely using trial and error.

$10 y^{2} - 53 y - 11$

Answer

(5 y + 1) (2 y - 11)

6.R-371 §6.R source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error

In the following exercises, factor completely using trial and error.

$−81 a^{2} +$ $153 a +$ $18$

Answer

−9 (9 a + 1) (a - 2)

6.R-373 §6.R source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ using the ‘ac’ Method

In the following exercises, factor.

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

Answer

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

6.R-375 §6.R source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ using the ‘ac’ Method

In the following exercises, factor.

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

Answer

Prime

6.R-377 §6.R source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ using the ‘ac’ Method

In the following exercises, factor.

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

Answer

3 (x + 4) (x - 3)

6.R-379 §6.R source ↗

Factor Trinomials of the Form $a x^{2} + b x + c$ using the ‘ac’ Method

In the following exercises, factor.

$18 a^{2} - 57 a - 21$

Answer

3 (2 a - 7) (3 a + 1)

6.R-381 §6.R source ↗

Factor using substitution

In the following exercises, factor using substitution.

$x^{4} - 13 x^{2} - 30$

Answer

(x^{2} - 15) (x^{2} + 2)

6.R-383 §6.R source ↗

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

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

Answer

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

6.R-385 §6.R source ↗

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

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

Answer

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

6.R-387 §6.R source ↗

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

$75 u^{4} -$ $30 u^{3} v +$ $3 u^{2} v^{2}$

Answer

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

6.R-389 §6.R source ↗

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

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

Answer

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

6.R-391 §6.R source ↗

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

$9 - 121 y^{2}$

Answer

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

6.R-393 §6.R source ↗

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

$169 n^{3} - n$

Answer

n (13 n + 1) (13 n - 1)

6.R-395 §6.R source ↗

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

$24 p^{2} + 54$

Answer

6 (4 p^{2} + 9)

6.R-397 §6.R source ↗

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

$16 z^{4} - 1$

Answer

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

6.R-399 §6.R source ↗

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

$a^{2} + 6 a + 9 - 9 b^{2}$

Answer

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

6.R-401 §6.R source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

$a^{3} - 125$

Answer

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

6.R-403 §6.R source ↗

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

$2 m^{3} + 54$

Answer

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

6.R-405 §6.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)

6.R-407 §6.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}

6.R-409 §6.R source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$5 u^{4} - 45 u^{2}$

Answer

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

6.R-411 §6.R source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$64 j^{2} + 225$

Answer

prime

6.R-413 §6.R source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$b^{3} - 64$

Answer

(b - 4) (b^{2} + 4 b + 16)

6.R-415 §6.R source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$2 b^{2} -$ $2 b c +$ $5 c b -$ $5 c^{2}$

Answer

(2 b + 5 c) (b - c)

6.R-417 §6.R source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$5 q^{2} - 15 q - 90$

Answer

5 (q + 3) (q - 6)

6.R-419 §6.R 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)

6.R-421 §6.R source ↗

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$16 x^{2} -$ $24 x y +$ $9 y^{2} -$ $64$

Answer

(4 x - 3 y + 8) (4 x - 3 y - 8)

6.R-423 §6.R source ↗

Use the Zero Product Property

In the following exercises, solve.

$(5 b + 1) (6 b + 1) =$ $0$

Answer

b =

−1 / 5,

b =

−1 / 6

6.R-425 §6.R source ↗

Use the Zero Product Property

In the following exercises, solve.

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

Answer

x = 1 / 2

6.R-427 §6.R source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

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

Answer

x = −4, x = −5

6.R-429 §6.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

6.R-431 §6.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}

6.R-433 §6.R source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

$(x + 6) (x - 3) =$ $−8$

Answer

x = 2, x = −5

6.R-435 §6.R source ↗

Solve Quadratic Equations by Factoring

In the following exercises, solve.

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

Answer

p = 0, p = ¾

6.R-437 §6.R source ↗

Solve Equations with Polynomial Functions

In the following exercises, solve.

For the function, $f (x) =$ $x^{2} +$ $11 x +$ $20,$ ⓐ find when $f (x) = −8$ ⓑ Use this information to find two points that lie on the graph of the function.

Answer

ⓐ

x = −7

or

x = −4

ⓑ

(−7, −8)

(−4, −8)

6.R-439 §6.R source ↗

In each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.

$f (x) =$ $64 x^{2} -$ $49$

Answer

ⓐ

x = \frac{7}{8}

or

x = - \frac{7}{8}

ⓑ

(\frac{7}{8}, 0),

(- \frac{7}{8}, 0)

ⓒ

(0, −49)

6.R-441 §6.R source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

The product of two consecutive odd numbers is 399. Find the numbers.

Answer

The numbers are

−21

and

−19

or 19 and 21.

6.R-443 §6.R source ↗

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

A ladder leans against the wall of a building. The length of the ladder is 9 feet longer than the distance of the bottom of the ladder from the building. The distance of the top of the ladder reaches up the side of the building is 7 feet longer than the distance of the bottom of the ladder from the building. Find the lengths of all three sides of the triangle formed by the ladder leaning against the building.

Answer

The lengths are 8, 15, and 17 ft.

6.T-445 §6.T source ↗

In the following exercises, factor completely.

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

Answer

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

6.T-447 §6.T source ↗

In the following exercises, factor completely.

$x^{2} + 13 x + 36$

Answer

(x + 4) (x + 9)

6.T-449 §6.T source ↗

In the following exercises, factor completely.

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

Answer

(x - 8) (y + 7)

6.T-451 §6.T source ↗

In the following exercises, factor completely.

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

Answer

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

6.T-453 §6.T source ↗

In the following exercises, factor completely.

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

Answer

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

6.T-455 §6.T source ↗

In the following exercises, factor completely.

$x^{3} + 125$

Answer

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

6.T-457 §6.T source ↗

In the following exercises, factor completely.

$6 x^{4} - 19 x^{2} + 15$

Answer

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

6.T-459 §6.T source ↗

In the following exercises, solve

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

Answer

a = 4 / 5, a = −6

6.T-461 §6.T source ↗

In the following exercises, solve

The area of a rectangular place mat is 168 square inches. Its length is two inches longer than the width. Find the length and width of the placemat.

Answer

The width is 12 inches and the length is 14 inches.

6.T-463 §6.T source ↗

In the following exercises, solve

For the function, $f (x) =$ $x^{2} -$ $7 x +$ $5,$ ⓐ find when $f (x) = −7$ ⓑ Use this information to find two points that lie on the graph of the function.

Answer

ⓐ

x = 3

or

x = 4

ⓑ

(3, −7)

(4, −7)