Content check — Intermediate Algebra, Chapter 5: Polynomials and Polynomial Functions

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 5: Polynomials and Polynomial Functions

362 problems · 0 figures

Be Prepared 5.1 §5.1 source ↗

Simplify:

3 x^{2} +

3 x +

1 +

8 x^{2} +

5 x +

5 .

If you missed this problem, review Example 1.7.

Example 1.7 in the book ↗

Answer

11 x^{2} + 8 x + 6

Be Prepared 5.2 §5.1 source ↗

Subtract:

(5 n + 8) -

(2 n - 1) .

If you missed this problem, review Example 1.5.

Example 1.5 in the book ↗

Answer

3 n + 9

Be Prepared 5.3 §5.1 source ↗

Evaluate:

4 x y^{2}

when

x = −2

and

y = 5 .

If you missed this problem, review Example 1.21.

Example 1.21 in the book ↗

Answer

- 200

Try It 5.1 §5.1 source ↗

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

ⓐ $−5$ ⓑ $8 y^{3} - 7 y^{2} - y - 3$ ⓒ $−3 x^{2} y -$ $5 x y +$ $9 x y^{3}$ ⓓ $81 m^{2} - 4 n^{2}$ ⓔ $−3 x^{6} y^{3} z$

Answer

ⓐ monomial, 0
ⓑ polynomial, 3 ⓒ trinomial, 4
ⓓ binomial, 2 ⓔ monomial, 10

Try It 5.2 §5.1 source ↗

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

ⓐ $64 k^{3} - 8$ ⓑ $9 m^{3} + 4 m^{2} - 2$ ⓒ $\frac{5}{6}$ ⓓ $8 a^{4} -$ $7 a^{3} b -$ $6 a^{2} b^{2} -$ $4 a b^{3} +$ $7 b^{4}$ ⓔ $− p^{4} q^{3}$

Answer

ⓐ binomial, 3 ⓑ trinomial, 3 ⓒ monomial, 0 ⓓ polynomial, 4 ⓔ monomial, 7

Try It 5.3 §5.1 source ↗

Add or subtract: ⓐ

12 q^{2} + 9 q^{2}

ⓑ

8 m n^{3} -

(−5 m n^{3}) .

Answer

ⓐ

21 q^{2}

ⓑ

13 m n^{3}

Try It 5.4 §5.1 source ↗

Add or subtract: ⓐ

−15 c^{2} + 8 c^{2}

ⓑ

−15 y^{2} z^{3} -

(−5 y^{2} z^{3}) .

Answer

ⓐ

−7 c^{2}

ⓑ

−10 y^{2} z^{3}

Try It 5.5 §5.1 source ↗

Add: ⓐ

8 y^{2} + 3 z^{2} - 3 y^{2}

ⓑ

m^{2} n^{2} -

8 m^{2} +

4 n^{2} .

Answer

ⓐ

5 y^{2} + 3 z^{2}

ⓑ

m^{2} n^{2} -

8 m^{2} +

4 n^{2}

Try It 5.6 §5.1 source ↗

Add: ⓐ

3 m^{2} + n^{2} - 7 m^{2}

ⓑ

p q^{2} - 6 p - 5 q^{2} .

Answer

ⓐ

−4 m^{2} + n^{2}

ⓑ

p q^{2} - 6 p - 5 q^{2}

Try It 5.7 §5.1 source ↗

Find the sum:

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

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

Answer

8 x^{2} - 11 x + 8

Try It 5.8 §5.1 source ↗

Find the sum:

(14 y^{2} + 6 y - 4) +

(3 y^{2} + 8 y + 5) .

Answer

17 y^{2} + 14 y + 1

Try It 5.9 §5.1 source ↗

Find the difference:

(8 x^{2} + 3 x - 19) -

(7 x^{2} - 14) .

Answer

x^{2} + 3 x - 5

Try It 5.10 §5.1 source ↗

Find the difference:

(9 b^{2} - 5 b - 4) -

(3 b^{2} - 5 b - 7) .

Answer

6 b^{2} + 3

Try It 5.11 §5.1 source ↗

Subtract

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

from

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

Answer

−5 a b + 7 b^{2}

Try It 5.12 §5.1 source ↗

Subtract

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

from

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

Answer

7 m n + 4 n^{2}

Try It 5.13 §5.1 source ↗

Find the sum:

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

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

Answer

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

Try It 5.14 §5.1 source ↗

Find the sum:

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

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

Answer

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

Try It 5.15 §5.1 source ↗

Simplify:

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

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

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

Answer

x^{3} - y^{3}

Try It 5.16 §5.1 source ↗

Simplify:

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

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

(p^{2} q + p q^{2}) .

Answer

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

Try It 5.17 §5.1 source ↗

For the function

f (x) =

3 x^{2} +

2 x -

15,

find ⓐ

f (3)

ⓑ

f (−5)

ⓒ

f (0) .

Answer

ⓐ 18 ⓑ 50 ⓒ

−15

Try It 5.18 §5.1 source ↗

For the function

g (x) =

5 x^{2} -

x -

4,

find ⓐ

g (−2)

ⓑ

g (−1)

ⓒ

g (0) .

Answer

ⓐ 18 ⓑ 2 ⓒ

−4

Try It 5.19 §5.1 source ↗

The polynomial function

h (t) =

−16 t^{2} +

150

gives the height of a stone t seconds after it is dropped from a 150-foot tall cliff. Find the height after

t = 0

seconds (the initial height of the object).

Answer

The height is

150

feet.

Try It 5.20 §5.1 source ↗

The polynomial function

h (t) =

−16 t^{2} +

175

gives the height of a ball t seconds after it is dropped from a 175-foot tall bridge. Find the height after

t = 3

seconds.

Answer

The height is 31 feet.

Try It 5.21 §5.1 source ↗

For functions

f (x) =

2 x^{2} -

4 x +

3

and

g (x) =

x^{2} -

2 x -

6,

find: ⓐ

(f + g) (x)

ⓑ

(f + g) (3)

ⓒ

(f - g) (x)

ⓓ

(f - g) (−2) .

Answer

ⓐ

(f + g) (x) =

3 x^{2} -

6 x -

3

ⓑ

(f + g) (3) = 6

ⓒ

(f - g) (x) =

x^{2} -

2 x +

9

ⓓ

(f - g) (−2) =

17

Try It 5.22 §5.1 source ↗

For functions

f (x) =

5 x^{2} -

4 x -

1

and

g (x) =

x^{2} +

3 x +

8,

find ⓐ

(f + g) (x)

ⓑ

(f + g) (3)

ⓒ

(f - g) (x)

ⓓ

(f - g) (−2) .

Answer

ⓐ

(f + g) (x) =

6 x^{2} -

x +

7

ⓑ

(f + g) (3) = 58

ⓒ

(f - g) (x) =

4 x^{2} -

7 x -

9

ⓓ

(f - g) (−2) =

21

5.1-1 §5.1 source ↗

Determine the Type of Polynomials

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.

ⓐ $47 x^{5} -$ $17 x^{2} y^{3} +$ $y^{2}$
ⓑ $5 c^{3} +$ $11 c^{2} -$ $c -$ $8$
ⓒ $\frac{5}{9} a b + \frac{1}{3} b$
ⓓ 4
ⓔ $4 p q + 17$

Answer

ⓐ trinomial, 5 ⓑ polynomial, 3 ⓒ binomial, 2 ⓓ monomial, 0
ⓔ binomial, 2

5.1-3 §5.1 source ↗

Determine the Type of Polynomials

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.

ⓐ $8 y - 5 x$
ⓑ $y^{2} - 5 y z - 6 z^{2}$
ⓒ $y^{3} -$ $8 y^{2} +$ $2 y -$ $16$
ⓓ $81 a b^{4} -$ $24 a^{2} b^{2} +$ $3 b$
ⓔ $−18$

Answer

ⓐ binomial, 1ⓑ trinomial, 2
ⓒ polynomial, 3ⓓ trinomial, 5
ⓔ monomial, 0

5.1-5 §5.1 source ↗

Determine the Type of Polynomials

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.

ⓐ $5 a^{2} +$ $12 a b -$ $7 b^{2}$
ⓑ $18 x y^{2} z$
ⓒ $5 x + 2$
ⓓ $y^{3} -$ $8 y^{2} +$ $2 y -$ $16$
ⓔ $−24$

Answer

ⓐ

trinomial, 2

ⓑ

monomial, 4

ⓒ

binomial, 1

ⓓ

polynomial, 3

ⓔ

monomial, 0

5.1-7 §5.1 source ↗

Determine the Type of Polynomials

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.

ⓐ $14 s - 29 t$
ⓑ $z^{2} - 5 z - 6$
ⓒ $y^{3} -$ $8 y^{2} z +$ $2 y z^{2} -$ $16 z^{3}$
ⓓ $23 a b^{2} - 14$
ⓔ $−3$

Answer

ⓐ

binomial, 1

ⓑ

trinomial, 2

ⓒ

polynomial, 3

ⓓ

binomial, 3

ⓔ

monomial, 0

5.1-9 §5.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

ⓐ ${7x}^{2} + 5 x^{2}$
ⓑ $4a - 9 a$

Answer

ⓐ

{12x}^{2}

ⓑ

− 5a

5.1-11 §5.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

ⓐ $−12 w + 18 w$
ⓑ $7 x^{2} y -$ $(−12 x^{2} y)$

Answer

ⓐ

6 w

ⓑ

19 x^{2} y

5.1-13 §5.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

${7x}^{2} +$ $5 x^{2} +$ $4a -$ $9 a$

Answer

{12x}^{2} - 5a

5.1-15 §5.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

$−12 w +$ $18 w +$ $7 x^{2} y -$ $(−12 x^{2} y)$

Answer

6 w + 19 x^{2} y

5.1-17 §5.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

ⓐ $−5 b - 17 b$
ⓑ $3 x y -$ $(−8 x y) +$ $5 x y$

Answer

ⓐ

−22 b

ⓑ

16 x y

5.1-19 §5.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

ⓐ $12 a + 5 b - 22 a$
ⓑ $p q^{2} - 4 p - 3 q^{2}$

Answer

ⓐ

−10 a + 5 b

ⓑ

p q^{2} - 4 p - 3 q^{2}

5.1-21 §5.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

ⓐ $2 a^{2} + b^{2} - 6 a^{2}$
ⓑ $x^{2} y - 3 x + 7 x y^{2}$

Answer

ⓐ

−4 a^{2} + b^{2}

ⓑ

x^{2} y - 3 x + 7 x y^{2}

5.1-23 §5.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

ⓐ $x y^{2} - 5 x - 5 y^{2}$
ⓑ $19y + 5 z$

Answer

ⓐ

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

ⓑ

19 y + 5 z

5.1-25 §5.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

$14x -$ $3 y -$ $13 x +$ $a^{2} b -$ $4 a -$ $5 a b^{2}$

Answer

x -

3 y +

a^{2} b -

4 a -

5 a b^{2}

5.1-27 §5.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

$5 u^{2} +$ $4 v^{2} -$ $6 u^{2} +$ $12a +$ $8 b$

Answer

− u^{2} +

4 v^{2} +

12a +

8 b

5.1-29 §5.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

Add: $4 a, −3 b, −8 a$

Answer

− 4a - 3 b

5.1-31 §5.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

Subtract $5 x^{6}$ from $−12 x^{6}$

Answer

−17 x^{6}

5.1-33 §5.1 source ↗

In the following exercises, add the polynomials.

$(5 y^{2} + 12 y + 4) +$ $(6 y^{2} - 8 y + 7)$

Answer

11 y^{2} + 4 y + 11

5.1-35 §5.1 source ↗

In the following exercises, add the polynomials.

$(x^{2} + 6 x + 8) +$ $(−4 x^{2} + 11 x - 9)$

Answer

−3 x^{2} + 17 x - 1

5.1-37 §5.1 source ↗

In the following exercises, add the polynomials.

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

Answer

11 x^{2} - 5 x + 5

5.1-39 §5.1 source ↗

In the following exercises, add the polynomials.

$(5 a^{2} + 8) +$ $(a^{2} - 4 a - 9)$

Answer

6 a^{2} - 4 a - 1

5.1-41 §5.1 source ↗

In the following exercises, subtract the polynomials.

$(4 m^{2} - 6 m - 3) -$ $(2 m^{2} + m - 7)$

Answer

2 m^{2} - 7 m + 4

5.1-43 §5.1 source ↗

In the following exercises, subtract the polynomials.

$(a^{2} + 8 a + 5) -$ $(a^{2} - 3 a + 2)$

Answer

11 a + 3

5.1-45 §5.1 source ↗

In the following exercises, subtract the polynomials.

$(12 s^{2} - 15 s) -$ $(s - 9)$

Answer

12 s^{2} - 16 s + 9

5.1-47 §5.1 source ↗

In the following exercises, subtract the polynomials.

Subtract $(9 x^{2} + 2)$ from $(12 x^{2} - x + 6)$

Answer

3 x^{2} - x + 4

5.1-49 §5.1 source ↗

In the following exercises, subtract the polynomials.

Subtract $(7 w^{2} - 4 w + 2)$ from $(8 w^{2} - w + 6)$

Answer

w^{2} + 3 w + 4

5.1-51 §5.1 source ↗

In the following exercises, find the difference of the polynomials.

Find the difference of $(w^{2} + w - 42)$ and $(w^{2} - 10 w + 24)$

Answer

11 w - 66

5.1-53 §5.1 source ↗

In the following exercises, add the polynomials.

$(7 x^{2} - 2 x y + 6 y^{2}) +$ $(3 x^{2} - 5 x y)$

Answer

10 x^{2} -

7 x y +

6 y^{2}

5.1-55 §5.1 source ↗

In the following exercises, add the polynomials.

$(7 m^{2} + m n - 8 n^{2}) +$ $(3 m^{2} + 2 m n)$

Answer

10 m^{2} +

3 m n -

8 n^{2}

5.1-57 §5.1 source ↗

In the following exercises, add or subtract the polynomials.

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

Answer

−3 a b + 3 b^{2}

5.1-59 §5.1 source ↗

In the following exercises, add or subtract the polynomials.

$(p^{3} - 3 p^{2} q) +$ $(2 p q^{2} + 4 q^{3}) -$ $(3 p^{2} q + p q^{2})$

Answer

p^{3} -

6 p^{2} q +

p q^{2} +

4 q^{3}

5.1-61 §5.1 source ↗

In the following exercises, add or subtract the polynomials.

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

Answer

x^{3} +

2 x^{2} y -

5 x y^{2} +

y^{3}

5.1-63 §5.1 source ↗

Evaluate a Polynomial Function for a Given Value

In the following exercises, find the function values for each polynomial function.

For the function $f (x) =$ $8 x^{2} -$ $3 x +$ $2,$ find:
ⓐ $f (5)$ ⓑ $f (−2)$ ⓒ $f (0)$

Answer

ⓐ 187 ⓑ 40 ⓒ 2

5.1-65 §5.1 source ↗

Evaluate a Polynomial Function for a Given Value

In the following exercises, find the function values for each polynomial function.

For the function $g (x) = 4 - 36 x,$ find:
ⓐ $g (3)$ ⓑ $g (0)$ ⓒ $g (−1)$

Answer

ⓐ

−104

ⓑ 4 ⓒ 40

5.1-67 §5.1 source ↗

In the following exercises, find the height for each polynomial function.

A painter drops a brush from a platform 75 feet high. The polynomial function $h (t) =$ $−16 t^{2} +$ $75$ gives the height of the brush t seconds after it was dropped. Find the height after $t = 2$ seconds.

Answer

The height is 11 feet.

5.1-69 §5.1 source ↗

In the following exercises, find the height for each polynomial function.

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial function $R (p) =$ $−4 p^{2} +$ $420 p .$ Find the revenue received when $p = 60$ dollars.

Answer

The revenue is $10,800.

5.1-71 §5.1 source ↗

In the following exercises, find the height for each polynomial function.

The polynomial $C (x) =$ $6 x^{2} +$ $90 x$ gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and height 6 feet. Find the cost of producing a box with $x = 4$ feet.

Answer

The cost is $456.

5.1-73 §5.1 source ↗

Add and Subtract Polynomial Functions

In each example, find ⓐ (f + g)(x) ⓑ (f + g)(2) ⓒ (f − g)(x) ⓓ (f − g)(−3).

$f (x) =$ $2 x^{2} -$ $4 x +$ $1$ and $g (x) =$ $5 x^{2} +$ $8 x +$ $3$

Answer

ⓐ

(f + g) (x) =

7 x^{2} +

4 x +

4

ⓑ

(f + g) (2) = 40

ⓒ

(f - g) (x) =

−3 x^{2} -

12 x -

2

ⓓ

(f - g) (−3) = 7

5.1-75 §5.1 source ↗

Add and Subtract Polynomial Functions

In each example, find ⓐ (f + g)(x) ⓑ (f + g)(2) ⓒ (f − g)(x) ⓓ (f − g)(−3).

$f (x) =$ $3 x^{3} -$ $x^{2} -$ $2 x +$ $3$ and $g (x) = 3 x^{3} - 7 x$

Answer

ⓐ

(f + g) (x) =

6 x^{3} -

x^{2} -

9 x +

3

ⓑ

(f + g) (2) = 29

ⓒ

(f - g) (x) =

− x^{2} +

5 x +

3

ⓓ

(f - g) (−3) =

−21

5.1-77 §5.1 source ↗

Add and Subtract Polynomial Functions

In each example, find ⓐ (f + g)(x) ⓑ (f + g)(2) ⓒ (f − g)(x) ⓓ (f − g)(−3).

Using your own words, explain the difference between a monomial, a binomial, and a trinomial.

Answer

Answers will vary.

5.1-79 §5.1 source ↗

Add and Subtract Polynomial Functions

In each example, find ⓐ (f + g)(x) ⓑ (f + g)(2) ⓒ (f − g)(x) ⓓ (f − g)(−3).

Ariana thinks the sum $6 y^{2} + 5 y^{4}$ is $11 y^{6} .$ What is wrong with her reasoning?

Answer

Answers will vary.

Be Prepared 5.4 §5.2 source ↗

Simplify:

(−2) (−2) (−2) .

If you missed this problem, review Example 1.19.

Example 1.19 in the book ↗

Answer

- 8

Be Prepared 5.5 §5.2 source ↗

Simplify:

\frac{8 x}{24 y} .

If you missed this problem, review Example 1.24.

Example 1.24 in the book ↗

Answer

\frac{x}{3 y}

Be Prepared 5.6 §5.2 source ↗

Name the decimal

(−2.6) (4.21) .

If you missed this problem, review Example 1.36.

Example 1.36 in the book ↗

Answer

- 10.946

Try It 5.23 §5.2 source ↗

Simplify each expression:

ⓐ $b^{9} \cdot b^{8}$ ⓑ $4^{2 x} \cdot 4^{x}$ ⓒ $3 p^{5} \cdot 4 p$ ⓓ $x^{6} \cdot x^{4} \cdot x^{8} .$

Answer

ⓐ

b^{17}

ⓑ

4^{3 x}

ⓒ

12 p^{6}

ⓓ

x^{18}

Try It 5.24 §5.2 source ↗

Simplify each expression:

ⓐ $x^{12} \cdot x^{4}$ ⓑ $10 \cdot 10^{x}$ ⓒ $2 z \cdot 6 z^{7}$ ⓓ $b^{5} \cdot b^{9} \cdot b^{5} .$

Answer

ⓐ

x^{16}

ⓑ

10^{x + 1}

ⓐ

12 z^{8}

ⓓ

b^{19}

Try It 5.25 §5.2 source ↗

Simplify each expression: ⓐ

\frac{x^{15}}{x^{10}}

ⓑ

\frac{6^{14}}{6^{5}}

ⓒ

\frac{x^{18}}{x^{22}}

ⓓ

\frac{12^{15}}{12^{30}} .

Answer

ⓐ

x^{5}

ⓑ

6^{9}

ⓒ

\frac{1}{x^{4}}

ⓓ

\frac{1}{12^{15}}

Try It 5.26 §5.2 source ↗

Simplify each expression: ⓐ

\frac{y^{43}}{y^{37}}

ⓑ

\frac{10^{15}}{10^{7}}

ⓒ

\frac{m^{7}}{m^{15}}

ⓓ

\frac{9^{8}}{9^{19}} .

Answer

ⓐ

y^{6}

ⓑ

10^{8}

ⓒ

\frac{1}{m^{8}}

ⓓ

\frac{1}{9^{11}}

Try It 5.27 §5.2 source ↗

Simplify each expression: ⓐ

11^{0}

ⓑ

q^{0} .

Answer

ⓐ 1 ⓑ 1

Try It 5.28 §5.2 source ↗

Simplify each expression: ⓐ

23^{0}

ⓑ

r^{0} .

Answer

ⓐ 1 ⓑ 1

Try It 5.29 §5.2 source ↗

Simplify each expression: ⓐ

z^{−3}

ⓑ

10^{−7}

ⓒ

\frac{1}{p^{−8}}

ⓓ

\frac{1}{4^{−3}} .

Answer

ⓐ

\frac{1}{z^{3}}

ⓑ

\frac{1}{10^{7}}

ⓒ

p^{8}

ⓓ

64

Try It 5.30 §5.2 source ↗

Simplify each expression: ⓐ

n^{−2}

ⓑ

10^{−4}

ⓒ

\frac{1}{q^{−7}}

ⓓ

\frac{1}{2^{−4}} .

Answer

ⓐ

\frac{1}{n^{2}}

ⓑ

\frac{1}{10,000}

ⓒ

q^{7}

ⓓ

16

Try It 5.31 §5.2 source ↗

Simplify each expression: ⓐ

{(\frac{2}{3})}^{−4}

ⓑ

{(- \frac{m}{n})}^{−2} .

Answer

ⓐ

\frac{81}{16}

ⓑ

\frac{n^{2}}{m^{2}}

Try It 5.32 §5.2 source ↗

Simplify each expression: ⓐ

{(\frac{3}{5})}^{−3}

ⓑ

{(- \frac{a}{b})}^{−4} .

Answer

ⓐ

\frac{125}{27}

ⓑ

\frac{b^{4}}{a^{4}}

Try It 5.33 §5.2 source ↗

Simplify each expression:

ⓐ $z^{−4} \cdot z^{−5}$ ⓑ $(p^{6} q^{−2}) (p^{−9} q^{−1})$ ⓒ $(3 u^{−5} v^{7}) (−4 u^{4} v^{−2}) .$

Answer

ⓐ

\frac{1}{z^{9}}

ⓑ

\frac{1}{p^{3} q^{3}}

ⓒ

- \frac{12 v^{5}}{u}

Try It 5.34 §5.2 source ↗

Simplify each expression:

ⓐ $c^{−8} \cdot c^{−7}$ ⓑ $(r^{5} s^{−3}) (r^{−7} s^{−5})$ ⓒ $(−6 c^{−6} d^{4}) (−5 c^{−2} d^{−1}) .$

Answer

ⓐ

\frac{1}{c^{15}}

ⓑ

\frac{1}{r^{2} s^{8}}

ⓒ

\frac{30 d^{3}}{c^{8}}

Try It 5.35 §5.2 source ↗

Simplify each expression: ⓐ

{(b^{7})}^{5}

ⓑ

{(5^{4})}^{3}

ⓒ

{(a^{4})}^{5} {(a^{7})}^{4} .

Answer

ⓐ

b^{35}

ⓑ

5^{12}

ⓒ

a^{48}

Try It 5.36 §5.2 source ↗

Simplify each expression: ⓐ

{(z^{6})}^{9}

ⓑ

{(3^{7})}^{7}

ⓒ

{(q^{4})}^{5} {(q^{3})}^{3} .

Answer

ⓐ

z^{54}

ⓑ

3^{49}

ⓒ

q^{29}

Try It 5.37 §5.2 source ↗

Simplify each expression: ⓐ

{(2 w x)}^{5}

ⓑ

{(−11 p q^{3})}^{0}

ⓒ

{(2 b^{3})}^{−4}

ⓓ

{(8 a^{−4})}^{2} .

Answer

ⓐ

32 w^{5} x^{5}

ⓑ 1 ⓒ

\frac{1}{16 b^{12}}

ⓓ

\frac{64}{a^{8}}

Try It 5.38 §5.2 source ↗

Simplify each expression: ⓐ

{(−3 y)}^{3}

ⓑ

{(−8 m^{2} n^{3})}^{0}

ⓒ

{(−4 x^{4})}^{−2}

ⓓ

{(2 c^{−4})}^{3} .

Answer

ⓐ

−27 y^{3}

ⓑ 1 ⓒ

\frac{1}{16 x^{8}}

ⓓ

\frac{8}{c^{12}}

Try It 5.39 §5.2 source ↗

Simplify each expression:

ⓐ ${(\frac{p}{10})}^{4}$ ⓑ ${(\frac{m}{n})}^{−7}$ ⓒ ${(\frac{3 a b^{3}}{c^{2}})}^{4}$ ⓓ ${(\frac{3 x^{−2}}{y^{3}})}^{3} .$

Answer

ⓐ

\frac{p^{4}}{10000}

ⓑ

\frac{n^{7}}{m^{7}}

ⓒ

\frac{81 a^{4} b^{12}}{c^{8}}

ⓓ

\frac{27}{x^{6} y^{9}}

Try It 5.40 §5.2 source ↗

Simplify each expression:

ⓐ ${(\frac{−2}{q})}^{3}$ ⓑ ${(\frac{w}{x})}^{−4}$ ⓒ ${(\frac{x y^{3}}{3 z^{2}})}^{2}$ ⓓ ${(\frac{2 m^{−2}}{n^{−2}})}^{3} .$

Answer

ⓐ

\frac{−8}{q^{3}}

ⓑ

\frac{x^{4}}{w^{4}}

ⓒ

\frac{x^{2} y^{6}}{9 z^{4}}

ⓓ

\frac{8 n^{6}}{m^{6}}

Try It 5.41 §5.2 source ↗

Simplify each expression:

ⓐ ${(c^{4} d^{2})}^{5} {(3 c d^{5})}^{4}$ ⓑ $\frac{{(a^{−2})}^{3} {(a^{2})}^{4}}{{(a^{4})}^{5}}$ ⓒ ${(\frac{3 x y^{2}}{x^{2} y^{−3}})}^{2} {(\frac{9 x y^{−3}}{x^{3} y^{2}})}^{−1} .$

Answer

ⓐ

81 c^{24} d^{30}

ⓑ

\frac{1}{a^{18}}

ⓒ

y^{15}

Try It 5.42 §5.2 source ↗

Simplify each expression:

ⓐ ${(a^{3} b^{2})}^{6} {(4 a b^{3})}^{4}$ ⓑ $\frac{{(p^{−3})}^{4} {(p^{5})}^{3}}{{(p^{7})}^{6}}$ ⓒ ${(\frac{4 x^{3} y^{2}}{x^{2} y^{−1}})}^{2} {(\frac{8 x y^{−3}}{x^{2} y^{}})}^{−1} .$

Answer

ⓐ

256 a^{22} b^{24}

ⓑ

\frac{1}{p^{39}}

ⓒ

2 x^{3} y^{10}

Try It 5.43 §5.2 source ↗

Write in scientific notation: ⓐ 96,000 ⓑ 0.0078.

Answer

ⓐ

9.6 \times 10^{4}

ⓑ

7.8 \times 10^{−3}

Try It 5.44 §5.2 source ↗

Write in scientific notation: ⓐ 48,300 ⓑ 0.0129.

Answer

ⓐ

4.83 \times 10^{4}

ⓑ

1.29 \times 10^{−2}

Try It 5.45 §5.2 source ↗

Convert to decimal form: ⓐ

1.3 \times 10^{3}

ⓑ

−1.2 \times 10^{−4} .

Answer

ⓐ 1,300 ⓑ

−0.00012

Try It 5.46 §5.2 source ↗

Convert to decimal form: ⓐ

−9.5 \times 10^{4}

ⓑ

7.5 \times 10^{−2} .

Answer

ⓐ

−950,000

ⓑ 0.075

Try It 5.47 §5.2 source ↗

Multiply or divide as indicated. Write answers in decimal form:

ⓐ $(−3 \times 10^{5}) (2 \times 10^{−8})$ ⓑ $\frac{8 \times 10^{2}}{4 \times 10^{−2}} .$

Answer

ⓐ

−0.006

ⓑ 20,000

Try It 5.48 §5.2 source ↗

Multiply or divide as indicated. Write answers in decimal form:

ⓐ $(−3 \times 10^{−2}) (3 \times 10^{−1})$ ⓑ $\frac{8 \times 10^{4}}{2 \times 10^{−1}} .$

Answer

ⓐ

−0.009

ⓑ 400,000

5.2-81 §5.2 source ↗

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

ⓐ $d^{3} \cdot d^{6}$ ⓑ $4^{5 x} \cdot 4^{9 x}$ ⓒ $2 y \cdot 4 y^{3}$ ⓓ $w \cdot w^{2} \cdot w^{3}$

Answer

ⓐ

d^{9}

ⓑ

4^{14 x}

ⓒ

8 y^{4}

ⓓ

w^{6}

5.2-83 §5.2 source ↗

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

ⓐ $n^{19} \cdot n^{12}$ ⓑ $3^{x} \cdot 3^{6}$ ⓒ $7 w^{5} \cdot 8 w^{}$ ⓓ $a^{4} \cdot a^{3} \cdot a^{9}$

Answer

ⓐ

n^{31}

ⓑ

3^{x + 6}

ⓒ

56 w^{6}

ⓓ

a^{16}

5.2-85 §5.2 source ↗

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

$m^{x} \cdot m^{3}$

Answer

m^{x + 3}

5.2-87 §5.2 source ↗

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

$y^{a} \cdot y^{b}$

Answer

y^{a + b}

5.2-89 §5.2 source ↗

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

ⓐ $\frac{x^{18}}{x^{3}}$ ⓑ $\frac{5^{12}}{5^{3}}$ ⓒ $\frac{q^{18}}{q^{36}}$ ⓓ $\frac{10^{2}}{10^{3}}$

Answer

ⓐ

x^{15}

ⓑ

5^{9}

ⓒ

\frac{1}{q^{18}}

ⓓ

\frac{1}{10}

5.2-91 §5.2 source ↗

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

ⓐ $\frac{p^{21}}{p^{7}}$ ⓑ $\frac{4^{16}}{4^{4}}$ ⓒ $\frac{b}{b^{9}}$ ⓓ $\frac{4}{4^{6}}$

Answer

ⓐ

p^{14}

ⓑ

4^{12}

ⓒ

\frac{1}{b^{8}}

ⓓ

\frac{1}{4^{5}}

5.2-93 §5.2 source ↗

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

ⓐ $20^{0}$ ⓑ $b^{0}$

Answer

ⓐ 1 ⓑ 1

5.2-95 §5.2 source ↗

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

ⓐ $− 27^{0}$ ⓑ $− (27^{0})$

Answer

ⓐ

−1

ⓑ

−1

5.2-97 §5.2 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

ⓐ $a^{−2}$ ⓑ $10^{−3}$ ⓒ $\frac{1}{c^{−5}}$ ⓓ $\frac{1}{3^{−2}}$

Answer

ⓐ

\frac{1}{a^{2}}

ⓑ

\frac{1}{1000}

ⓒ

c^{5}

ⓓ

9

5.2-99 §5.2 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

ⓐ $r^{−3}$ ⓑ $10^{−5}$ ⓒ $\frac{1}{q^{−10}}$ ⓓ $\frac{1}{10^{−3}}$

Answer

ⓐ

\frac{1}{r^{3}}

ⓑ

\frac{1}{100,000}

ⓒ

q^{10}

ⓓ

1,000

5.2-101 §5.2 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

ⓐ ${(\frac{5}{8})}^{−2}$ ⓑ ${(- \frac{b}{a})}^{−2}$

Answer

ⓐ

\frac{64}{25}

ⓑ

\frac{a^{2}}{b^{2}}

5.2-103 §5.2 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

ⓐ ${(\frac{4}{9})}^{−3}$ ⓑ ${(- \frac{u}{v})}^{−5}$

Answer

ⓐ

\frac{729}{64}

ⓑ

- \frac{v^{5}}{u^{5}}

5.2-105 §5.2 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

ⓐ ${(−5)}^{−2}$ ⓑ $− 5^{−2}$ ⓒ ${(- \frac{1}{5})}^{−2}$ ⓓ $− {(\frac{1}{5})}^{−2}$

Answer

ⓐ

\frac{1}{25}

ⓑ

- \frac{1}{25}

ⓒ

25

ⓓ

−25

5.2-107 §5.2 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

ⓐ $3 \cdot 5^{−1}$ ⓑ ${(3 \cdot 5)}^{−1}$

Answer

ⓐ

\frac{3}{5}

ⓑ

\frac{1}{15}

5.2-109 §5.2 source ↗

In the following exercises, simplify each expression using the Product Property.

ⓐ $b^{4} b^{−8}$ ⓑ $(w^{4} x^{−5}) (w^{−2} x^{−4})$ ⓒ $(−6 c^{−3} d^{9}) (2 c^{4} d^{−5})$

Answer

ⓐ

\frac{1}{b^{4}}

ⓑ

\frac{w^{2}}{x^{9}}

ⓒ

−12 c d^{4}

5.2-111 §5.2 source ↗

In the following exercises, simplify each expression using the Product Property.

ⓐ $a^{3} \cdot a^{−3}$ ⓑ $(u v^{−2}) (u^{−5} v^{−3})$ ⓒ $(−4 r^{−2} s^{−8}) (9 r^{4} s^{3})$

Answer

ⓐ 1 ⓑ

\frac{1}{u^{4} v^{5}}

ⓒ

\frac{- 36 r^{2}}{s^{5}}

5.2-113 §5.2 source ↗

In the following exercises, simplify each expression using the Product Property.

$p^{5} \cdot p^{−2} \cdot p^{−4}$

Answer

\frac{1}{p}

5.2-115 §5.2 source ↗

In the following exercises, simplify each expression using the Power Property.

ⓐ ${(m^{4})}^{2}$ ⓑ ${(10^{3})}^{6}$ ⓒ ${(x^{3})}^{−4}$

Answer

ⓐ

m^{8}

ⓑ

10^{18}

ⓒ

\frac{1}{x^{12}}

5.2-117 §5.2 source ↗

In the following exercises, simplify each expression using the Power Property.

ⓐ ${(y^{3})}^{x}$ ⓑ ${(5^{x})}^{y}$ ⓒ ${(q^{6})}^{−8}$

Answer

ⓐ

y^{3 x}

ⓑ

5^{x y}

ⓒ

\frac{1}{q^{48}}

5.2-119 §5.2 source ↗

In the following exercises, simplify each expression using the Product to a Power Property.

ⓐ ${(−3 x y)}^{2}$ ⓑ ${(6 a)}^{0}$ ⓒ ${(5 x^{2})}^{−2}$ ⓓ ${(−4 y^{−3})}^{2}$

Answer

ⓐ

9 x^{2} y^{2}

ⓑ 1 ⓒ

\frac{1}{25 x^{4}}

ⓓ

\frac{16}{y^{6}}

5.2-121 §5.2 source ↗

In the following exercises, simplify each expression using the Product to a Power Property.

ⓐ ${(−5 a b)}^{3}$ ⓑ ${(−4 p q)}^{0}$ ⓒ ${(−6 x^{3})}^{−2}$ ⓓ ${(3 y^{−4})}^{2}$

Answer

ⓐ

−125 a^{3} b^{3}

ⓑ 1 ⓒ

\frac{1}{36 x^{6}}

ⓓ

\frac{9}{y^{8}}

5.2-123 §5.2 source ↗

In the following exercises, simplify each expression using the Quotient to a Power Property.

ⓐ ${(\frac{p}{2})}^{5}$ ⓑ ${(\frac{x}{y})}^{−6}$ ⓒ ${(\frac{2 x y^{2}}{z})}^{3}$ ⓓ ${(\frac{4 p^{−3}}{q^{2}})}^{2}$

Answer

ⓐ

\frac{p^{5}}{32}

ⓑ

\frac{y^{6}}{x^{6}}

ⓒ

\frac{8 x^{3} y^{6}}{z^{3}}

ⓓ

\frac{16}{p^{6} q^{4}}

5.2-125 §5.2 source ↗

In the following exercises, simplify each expression using the Quotient to a Power Property.

ⓐ ${(\frac{a}{3 b})}^{4}$ ⓑ ${(\frac{5}{4 m})}^{−2}$ ⓒ ${(\frac{3 a^{−2} b^{3}}{c^{2}})}^{−2}$ ⓓ ${(\frac{p^{−1} q^{4}}{r^{−4}})}^{2}$

Answer

ⓐ

\frac{a^{4}}{81 b^{4}}

ⓑ

\frac{16 m^{2}}{25}

ⓒ

\frac{a^{4} c^{4}}{9 b^{6}}

ⓓ

\frac{q^{8} r^{8}}{p^{2}}

5.2-127 §5.2 source ↗

In the following exercises, simplify each expression by applying several properties.

ⓐ ${(5 t^{2})}^{3} {(3 t)}^{2}$ ⓑ $\frac{{(t^{2})}^{5} {(t^{−4})}^{2}}{{(t^{3})}^{7}}$ ⓒ ${(\frac{2 x y^{2}}{x^{3} y^{−2}})}^{2} {(\frac{12 x y^{3}}{x^{3} y^{−1}})}^{−1}$

Answer

ⓐ

1125 t^{8}

ⓑ

\frac{1}{t^{19}}

ⓒ

\frac{y^{4}}{3 x^{2}}

5.2-129 §5.2 source ↗

In the following exercises, simplify each expression by applying several properties.

ⓐ ${(m^{2} n)}^{2} {(2 m n^{5})}^{4}$ ⓑ $\frac{{(−2 p^{−2})}^{4} {(3 p^{4})}^{2}}{{(−6 p^{3})}^{2}}$

Answer

ⓐ

16 m^{8} n^{22}

ⓑ

\frac{4}{p^{6}}

5.2-131 §5.2 source ↗

In the following exercises, simplify each expression.

ⓐ $7 n^{−1}$ ⓑ ${(7 n)}^{−1}$ ⓒ ${(−7 n)}^{−1}$

Answer

ⓐ

\frac{7}{n}

ⓑ

\frac{1}{7 n}

ⓒ

- \frac{1}{7 n}

5.2-133 §5.2 source ↗

In the following exercises, simplify each expression.

ⓐ ${(3 p)}^{−2}$ ⓑ $3 p^{−2}$ ⓒ $−3 p^{−2}$

Answer

ⓐ

\frac{1}{9 p^{2}}

ⓑ

\frac{3}{p^{2}}

ⓒ

\frac{−3}{p^{2}}

5.2-135 §5.2 source ↗

In the following exercises, simplify each expression.

${(x^{2})}^{4} \cdot {(x^{3})}^{2}$

Answer

x^{14}

5.2-137 §5.2 source ↗

In the following exercises, simplify each expression.

${(a^{2})}^{6} \cdot {(a^{3})}^{8}$

Answer

a^{36}

5.2-139 §5.2 source ↗

In the following exercises, simplify each expression.

${(2 m^{6})}^{3}$

Answer

8 m^{18}

5.2-141 §5.2 source ↗

In the following exercises, simplify each expression.

${(10 x^{2} y)}^{3}$

Answer

1,000 x^{6} y^{3}

5.2-143 §5.2 source ↗

In the following exercises, simplify each expression.

${(−2 a^{3} b^{2})}^{4}$

Answer

16 a^{12} b^{8}

5.2-145 §5.2 source ↗

In the following exercises, simplify each expression.

${(\frac{2}{3} x^{2} y)}^{3}$

Answer

\frac{8}{27} x^{6} y^{3}

5.2-147 §5.2 source ↗

In the following exercises, simplify each expression.

${(8 a^{3})}^{2} {(2 a)}^{4}$

Answer

1,024 a^{10}

5.2-149 §5.2 source ↗

In the following exercises, simplify each expression.

${(10 p^{4})}^{3} {(5 p^{6})}^{2}$

Answer

25,000 p^{24}

5.2-151 §5.2 source ↗

In the following exercises, simplify each expression.

${(\frac{1}{2} x^{2} y^{3})}^{4} {(4 x^{5} y^{3})}^{2}$

Answer

x^{18} y^{18}

5.2-153 §5.2 source ↗

In the following exercises, simplify each expression.

${(3 m^{2} n)}^{2} {(2 m n^{5})}^{4}$

Answer

144 m^{8} n^{22}

5.2-155 §5.2 source ↗

In the following exercises, simplify each expression.

ⓐ ${(3 x)}^{2} (5 x)$ ⓑ ${(2 y)}^{3} (6 y)$

Answer

ⓐ

45 x^{3}

ⓑ

48 y^{4}

5.2-157 §5.2 source ↗

In the following exercises, simplify each expression.

ⓐ ${(2 r^{−2})}^{3} {(4^{−1} r)}^{2}$ ⓑ ${(3 x^{−3})}^{3} {(3^{−1} x^{5})}^{4}$

Answer

ⓐ

\frac{1}{2 r^{4}}

ⓑ

\frac{1}{3} x^{11}

5.2-159 §5.2 source ↗

In the following exercises, simplify each expression.

${(\frac{j^{−2} j^{5}}{j^{4}})}^{3}$

Answer

\frac{1}{j^{3}}

5.2-161 §5.2 source ↗

In the following exercises, simplify each expression.

$\frac{{(−10 n^{−2})}^{3} {(4 n^{5})}^{2}}{{(2 n^{8})}^{2}}$

Answer

- \frac{4000}{n^{12}}

5.2-163 §5.2 source ↗

Use Scientific Notation

In the following exercises, write each number in scientific notation.

ⓐ 340,000 ⓑ 0.041

Answer

ⓐ

3.4 \times 10^{5}

ⓑ

4.1 \times 10^{−2}

5.2-165 §5.2 source ↗

Use Scientific Notation

In the following exercises, write each number in scientific notation.

ⓐ 1,290,000 ⓑ 0.00000103

Answer

ⓐ

1.29 \times 10^{6}

ⓑ

1.03 \times 10^{−6}

5.2-167 §5.2 source ↗

In the following exercises, convert each number to decimal form.

ⓐ $−8.3 \times 10^{2}$ ⓑ $3.8 \times 10^{−2}$

Answer

ⓐ

−830

ⓑ 0.038

5.2-169 §5.2 source ↗

In the following exercises, convert each number to decimal form.

ⓐ $1.6 \times 10^{10}$ ⓑ $8.43 \times 10^{−6}$

Answer

ⓐ 16,000,000,000
ⓑ 0.00000843

5.2-171 §5.2 source ↗

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

ⓐ $(2 \times 10^{2}) (1 \times 10^{−4})$ ⓑ $\frac{5 \times 10^{−2}}{1 \times 10^{−10}}$

Answer

ⓐ 0.02 ⓑ 500,000,000

5.2-173 §5.2 source ↗

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

ⓐ $(3.5 \times 10^{−4}) (1.6 \times 10^{−2})$ ⓑ $\frac{8 \times 10^{6}}{4 \times 10^{−1}}$

Answer

ⓐ 0.0000056 ⓑ 20,000,000

5.2-175 §5.2 source ↗

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

Jennifer thinks the quotient $\frac{a^{24}}{a^{6}}$ simplifies to $a^{4} .$ What is wrong with her reasoning?

Answer

Answers will vary.

5.2-177 §5.2 source ↗

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

Answer

Answers will vary.

Be Prepared 5.7 §5.3 source ↗

Distribute:

2 (x + 3) .

If you missed this problem, review Example 1.50.

Example 1.50 in the book ↗

Answer

2 x + 6

Be Prepared 5.8 §5.3 source ↗

Simplify: ⓐ

9^{2}

ⓑ

{(−9)}^{2}

ⓒ

− 9^{2} .

If you missed this problem, review Example 1.19.

Example 1.19 in the book ↗

Answer

ⓐ

81

; ⓑ

81

; ⓒ

- 81

Be Prepared 5.9 §5.3 source ↗

Evaluate:

2 x^{2} - 5 x + 3

for

x = −2 .

If you missed this problem, review Example 1.21.

Example 1.21 in the book ↗

Answer

21

Try It 5.49 §5.3 source ↗

Multiply: ⓐ

(5 y^{7}) (−7 y^{4})

ⓑ

(\frac{2}{5} a^{4} b^{3}) (15 a b^{3}) .

Answer

ⓐ

−35 y^{11}

ⓑ

6 a^{5} b^{6}

Try It 5.50 §5.3 source ↗

Multiply: ⓐ

(−6 b^{4}) (−9 b^{5})

ⓑ

(\frac{2}{3} r^{5} s) (12 r^{6} s^{7}) .

Answer

ⓐ

54 b^{9}

ⓑ

8 r^{11} s^{8}

Try It 5.51 §5.3 source ↗

Multiply: ⓐ

−3 y (5 y^{2} + 8 y - 7)

ⓑ

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

Answer

ⓐ

−15 y^{3} -

24 y^{2} +

21 y

ⓑ

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

20 x^{3} y^{3} +

12 x^{2} y^{4}

Try It 5.52 §5.3 source ↗

Multiply: ⓐ

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

ⓑ

−6 a^{3} b (3 a^{2} - 2 a b + 6 b^{2}) .

Answer

ⓐ

8 x^{4} -

12 x^{3} +

20 x^{2}

ⓑ

−18 a^{5} b +

12 a^{4} b^{2} -

36 a^{3} b^{3}

Try It 5.53 §5.3 source ↗

Multiply: ⓐ

(x + 8) (x + 9)

ⓑ

(3 c + 4) (5 c - 2) .

Answer

ⓐ

x^{2} + 17 x + 72

ⓑ

15 c^{2} + 14 c - 8

Try It 5.54 §5.3 source ↗

Multiply: ⓐ

(5 x + 9) (4 x + 3)

ⓑ

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

Answer

ⓐ

20 x^{2} + 51 x + 27

ⓑ

30 y^{2} - 3 y - 6

Try It 5.55 §5.3 source ↗

Multiply: ⓐ

(x - 7) (x + 5)

ⓑ

(3 x + 7) (5 x - 2) .

Answer

ⓐ

x^{2} - 2 x - 35

ⓑ

15 x^{2} + 29 x - 14

Try It 5.56 §5.3 source ↗

Multiply: ⓐ

(b - 3) (b + 6)

ⓑ

(4 y + 5) (4 y - 10) .

Answer

ⓐ

b^{2} + 3 b - 18

ⓑ

16 y^{2} - 20 y - 50

Try It 5.57 §5.3 source ↗

Multiply: ⓐ

(x^{2} + 6) (x - 8)

ⓑ

(2 a b + 5) (4 a b - 4) .

Answer

ⓐ

x^{3} -

8 x^{2} +

6 x -

48

ⓑ

8 a^{2} b^{2} +

12 a b -

20

Try It 5.58 §5.3 source ↗

Multiply: ⓐ

(y^{2} + 7) (y - 9)

ⓑ

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

Answer

ⓐ

y^{3} -

9 y^{2} +

7 y -

63

ⓑ

8 x^{2} y^{2} +

2 x y -

15

Try It 5.59 §5.3 source ↗

Multiply using the Vertical Method:

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

Answer

15 m^{2} - 51 m + 42

Try It 5.60 §5.3 source ↗

Multiply using the Vertical Method:

(6 b - 5) (7 b - 3) .

Answer

42 b^{2} - 53 b + 15

Try It 5.61 §5.3 source ↗

Multiply

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

using ⓐ the Distributive Property and ⓑ the Vertical Method.

Answer

ⓐ

y^{3} -

8 y^{2} +

17 y -

6

ⓑ

y^{3} -

8 y^{2} +

17 y -

6

Try It 5.62 §5.3 source ↗

Multiply

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

using ⓐ the Distributive Property and ⓑ The Vertical Method.

Answer

ⓐ

2 x^{3} +

5 x^{2} -

7 x +

20

ⓑ

y^{3} -

8 y^{2} +

17 y -

6

Try It 5.63 §5.3 source ↗

Multiply: ⓐ

{(x + 9)}^{2}

ⓑ

{(2 c - d)}^{2} .

Answer

ⓐ

x^{2} + 18 x + 81

ⓑ

4 c^{2} - 4 c d + d^{2}

Try It 5.64 §5.3 source ↗

Multiply: ⓐ

{(y + 11)}^{2}

ⓑ

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

Answer

ⓐ

y^{2} + 22 y + 121

ⓑ

16 x^{2} -

40 x y +

25 y^{2}

Try It 5.65 §5.3 source ↗

Multiply: ⓐ

(6 x + 5) (6 x - 5)

ⓑ

(4 p - 7 q) (4 p + 7 q) .

Answer

ⓐ

36 x^{2} - 25

ⓑ

16 p^{2} - 49 q^{2}

Try It 5.66 §5.3 source ↗

Multiply: ⓐ

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

ⓑ

(3 x - y) (3 x + y) .

Answer

ⓐ

4 x^{2} - 49

ⓑ

9 x^{2} - y^{2}

Try It 5.67 §5.3 source ↗

Choose the appropriate pattern and use it to find the product:

ⓐ $(9 b - 2) (2 b + 9)$ ⓑ ${(9 p - 4)}^{2}$ ⓒ ${(7 y + 1)}^{2}$ ⓓ $(4 r - 3) (4 r + 3) .$

Answer

ⓐ FOIL;

18 b^{2} + 77 b - 18

ⓑ Binomial Squares;

81 p^{2} - 72 p + 16

49 y^{2} + 14 y + 1

ⓓ Product of Conjugates;

16 r^{2} - 9

Try It 5.68 §5.3 source ↗

Choose the appropriate pattern and use it to find the product:

ⓐ ${(6 x + 7)}^{2}$ ⓑ $(3 x - 4) (3 x + 4)$ ⓒ $(2 x - 5) (5 x - 2)$ ⓓ ${(6 n - 1)}^{2} .$

Answer

ⓐ Binomial Squares;

36 x^{2} + 84 x + 49

ⓑ Product of Conjugates;

9 x^{2} - 16

10 x^{2} - 29 x + 10

ⓓ Binomial Squares;

36 n^{2} - 12 n + 1

Try It 5.69 §5.3 source ↗

For functions

f (x) = x - 5

and

g (x) =

x^{2} -

2 x +

3,

find ⓐ

(f \cdot g) (x)

ⓑ

(f \cdot g) (2) .

Answer

ⓐ

(f \cdot g) (x) =

x^{3} -

7 x^{2} +

13 x -

15

ⓑ

(f \cdot g) (2) = −9

Try It 5.70 §5.3 source ↗

For functions

f (x) = x - 7

and

g (x) =

x^{2} +

8 x +

4,

find ⓐ

(f \cdot g) (x)

ⓑ

(f \cdot g) (2) .

Answer

ⓐ

(f \cdot g) (x) =

x^{3} +

x^{2} -

52 x -

28

ⓑ

(f \cdot g) (2) =

−120

5.3-179 §5.3 source ↗

Multiply Monomials

In the following exercises, multiply the monomials.

ⓐ $(−10 x^{5}) (−3 x^{3})$
ⓑ $(\frac{5}{8} x^{3} y) (24 x^{5} y)$

Answer

ⓐ

30 x^{8}

ⓑ

15 x^{8} y^{2}

5.3-181 §5.3 source ↗

Multiply Monomials

In the following exercises, multiply the monomials.

ⓐ $(−6 c^{4}) (−12 c)$
ⓑ $(\frac{3}{5} m^{3} n^{2}) (\frac{5}{9} m^{2} n^{3})$

Answer

ⓐ

72 c^{5}

ⓑ

\frac{1}{3} m^{5} n^{5}

5.3-183 §5.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

ⓐ $−5 t (t^{2} + 3 t - 18);$
ⓑ $9 r^{3} s (r^{2} - 3 r s + 5 s^{2})$

Answer

ⓐ

−5 t^{3} -

15 t^{2} +

90 t

ⓑ

9 s r^{5} -

27 s^{2} r^{4} +

45 s^{3} r^{3}

5.3-185 §5.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

ⓐ $−5 m (m^{2} + 3 m - 18)$
ⓑ $−3 x^{2} y^{2} (7 x^{2} + 10 x y - y^{2})$

Answer

ⓐ

−5 m^{3} -

15 m^{2} +

90 m

ⓑ

−21 x^{4} y^{2} -

30 x^{3} y^{3} +

3 x^{2} y^{4}

5.3-187 §5.3 source ↗

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using ⓐ the Distributive Property; ⓑ the FOIL method; ⓒ the Vertical Method.

$(y + 9) (y + 3)$

Answer

y^{2} + 12 y + 27

5.3-189 §5.3 source ↗

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using ⓐ the Distributive Property; ⓑ the FOIL method; ⓒ the Vertical Method.

$(7 q + 4) (3 q - 8)$

Answer

21 q^{2} - 44 q - 32

5.3-191 §5.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(y - 6) (y - 2)$

Answer

y^{2} - 8 y + 12

5.3-193 §5.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(6 p + 5) (p + 1)$

Answer

6 p^{2} + 11 p + 5

5.3-195 §5.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(m + 11) (m - 4)$

Answer

m^{2} + 7 m - 44

5.3-197 §5.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(3 r - 8) (11 r + 1)$

Answer

33 r^{2} - 85 r - 8

5.3-199 §5.3 source ↗

In the following exercises, multiply the binomials. Use any method.

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

Answer

y^{3} +

3 y^{2} -

4 y -

12

5.3-201 §5.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(2 x y + 3) (3 x y + 2)$

Answer

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

13 x y +

6

5.3-203 §5.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(y^{2} - 7) (y^{2} - 4)$

Answer

y^{4} - 11 y^{2} + 28

5.3-205 §5.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(3 r s - 7) (3 r s - 4)$

Answer

9 r^{2} s^{2} -

33 r s +

28

5.3-207 §5.3 source ↗

Multiply a Polynomial by a Polynomial

In the following exercises, multiply using ⓐ the Distributive Property; ⓑ the Vertical Method.

$(u + 4) (u^{2} + 3 u + 2)$

Answer

u^{3} +

7 u^{2} +

14 u +

8

5.3-209 §5.3 source ↗

Multiply a Polynomial by a Polynomial

In the following exercises, multiply using ⓐ the Distributive Property; ⓑ the Vertical Method.

$(a + 10) (3 a^{2} + a - 5)$

Answer

3 a^{3} +

31 a^{2} +

5 a -

50

5.3-211 §5.3 source ↗

Multiply a Polynomial by a Polynomial

In the following exercises, multiply using ⓐ the Distributive Property; ⓑ the Vertical Method.

$(2 a^{2} - 5 a + 10) (3 a^{2} + a - 5)$

Answer

6 a^{4} -

13 a^{3} +

15 a^{2} +

35 a -

50

5.3-213 §5.3 source ↗

Multiply Special Products

In the following exercises, multiply. Use either method.

$(p - 4) (p^{2} - 6 p + 9)$

Answer

p^{3} -

10 p^{2} +

33 p -

36

5.3-215 §5.3 source ↗

Multiply Special Products

In the following exercises, multiply. Use either method.

$(6 r + 1) (r^{2} - 7 r - 9)$

Answer

6 r^{3} -

41 r^{2} -

61 r -

9

5.3-217 §5.3 source ↗

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(q + 12)}^{2}$

Answer

q^{2} + 24 q + 144

5.3-219 §5.3 source ↗

In the following exercises, square each binomial using the Binomial Squares Pattern.

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

Answer

4 y^{2} -

12 y z +

9 z^{2}

5.3-221 §5.3 source ↗

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(x + \frac{2}{3})}^{2}$

Answer

x^{2} + \frac{4}{3} x + \frac{4}{9}

5.3-223 §5.3 source ↗

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(\frac{1}{8} x - \frac{1}{9} y)}^{2}$

Answer

\frac{1}{64} x^{2} -

\frac{1}{36} x y +

\frac{1}{81} y^{2}

5.3-225 §5.3 source ↗

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(5 u^{2} + 9)}^{2}$

Answer

25 u^{4} +

90 u^{2} +

81

5.3-227 §5.3 source ↗

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(8 p^{3} - 3)}^{2}$

Answer

64 p^{6} - 48 p^{3} + 9

5.3-229 §5.3 source ↗

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(8 j + 4) (8 j - 4)$

Answer

64 j^{2} - 16

5.3-231 §5.3 source ↗

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(9 c + 5) (9 c - 5)$

Answer

81 c^{2} - 25

5.3-233 §5.3 source ↗

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(7 w + 10 x) (7 w - 10 x)$

Answer

49 w^{2} - 100 x^{2}

5.3-235 §5.3 source ↗

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(p + \frac{4}{5} q) (p - \frac{4}{5} q)$

Answer

p^{2} - \frac{16}{25} q^{2}

5.3-237 §5.3 source ↗

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(x y - 9) (x y + 9)$

Answer

x^{2} y^{2} - 81

5.3-239 §5.3 source ↗

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(15 m^{2} - 8 n^{4}) (15 m^{2} + 8 n^{4})$

Answer

225 m^{4} - 64 n^{8}

5.3-241 §5.3 source ↗

In the following exercises, find each product.

${(t - 9)}^{2}$

Answer

t^{2} - 18 t + 81

5.3-243 §5.3 source ↗

In the following exercises, find each product.

$(2 x + y) (x - 2 y)$

Answer

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

5.3-245 §5.3 source ↗

In the following exercises, find each product.

$(3 p + 8) (3 p - 8)$

Answer

9 p^{2} - 64

5.3-247 §5.3 source ↗

In the following exercises, find each product.

${(k - 6)}^{2}$

Answer

k^{2} - 12 k + 36

5.3-249 §5.3 source ↗

In the following exercises, find each product.

$(x^{2} + 8 y) (8 x - y^{2})$

Answer

8 x^{3} -

x^{2} y^{2} +

64 x y -

8 y^{3}

5.3-251 §5.3 source ↗

In the following exercises, find each product.

${(y^{4} + 2 z)}^{2}$

Answer

y^{8} + 4 y^{4} z + 4 z^{2}

5.3-253 §5.3 source ↗

In the following exercises, find each product.

${(m^{3} - 8 n)}^{2}$

Answer

m^{6} -

16 m^{3} n +

64 n^{2}

5.3-255 §5.3 source ↗

In the following exercises, find each product.

$(r^{2} - s^{3}) (r^{3} + s^{2})$

Answer

r^{5} +

r^{2} s^{2} -

r^{3} s^{3} -

s^{5}

5.3-257 §5.3 source ↗

In the following exercises, find each product.

$(15 p - 4) +$ $(3 p - 5)$

Answer

18 p - 9

5.3-259 §5.3 source ↗

In the following exercises, find each product.

$(j^{2} - 8 j - 27) -$ $(j^{2} + 2 j - 12)$

Answer

−10 j - 15

5.3-261 §5.3 source ↗

In the following exercises, find each product.

$(\frac{1}{4} d^{5}) (36 d^{2})$

Answer

9 d^{7}

5.3-263 §5.3 source ↗

In the following exercises, find each product.

$(6 m^{4} n^{3}) (7 m n^{5})$

Answer

42 m^{5} n^{8}

5.3-265 §5.3 source ↗

In the following exercises, find each product.

$5 q^{3} (q^{2} - 2 q + 6)$

Answer

5 q^{5} -

10 q^{4} +

30 q^{3}

5.3-267 §5.3 source ↗

In the following exercises, find each product.

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

Answer

y^{3} - y^{2} - 2 y

5.3-269 §5.3 source ↗

In the following exercises, find each product.

$(6 k - 1) (k^{2} + 2 k - 4)$

Answer

6 k^{3} +

11 k^{2} -

26 k +

4

5.3-271 §5.3 source ↗

In the following exercises, find each product.

$(11 - b) (11 + b)$

Answer

121 - b^{2}

5.3-273 §5.3 source ↗

In the following exercises, find each product.

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

Answer

4 x^{4} - 9 y^{8}

5.3-275 §5.3 source ↗

In the following exercises, find each product.

${(3 d + 1)}^{2}$

Answer

9 d^{2} + 6 d + 1

5.3-277 §5.3 source ↗

In the following exercises, find each product.

${(3 z + \frac{1}{5})}^{2}$

Answer

9 z^{2} + \frac{6}{5} z + \frac{1}{25}

5.3-279 §5.3 source ↗

Multiply Polynomial Functions

For functions $f (x) = x - 1$ and $g (x) =$ $4 x^{2} +$ $3 x -$ $5,$ find ⓐ $(f \cdot g) (x)$ ⓑ $(f \cdot g) (−2)$

Answer

ⓐ

(f \cdot g) (x) =

4 x^{3} -

x^{2} -

8 x +

5

ⓑ

(f \cdot g) (−2) =

−15

5.3-281 §5.3 source ↗

Multiply Polynomial Functions

For functions $f (x) = 7 x - 8$ and $g (x) = 7 x + 8,$ find ⓐ $(f \cdot g) (x)$ ⓑ $(f \cdot g) (−2)$

Answer

ⓐ

(f \cdot g) (x) =

49 x^{2} -

64

ⓑ

(f \cdot g) (−2) =

132

5.3-283 §5.3 source ↗

Multiply Polynomial Functions

For functions $f (x) =$ $x^{2} +$ $4 x -$ $3$ and $g (x) =$ $x^{2} +$ $2 x +$ $4,$ find ⓐ $(f \cdot g) (x)$ ⓑ $(f \cdot g) (1)$

Answer

ⓐ

(f \cdot g) (x) =

x^{4} +

6 x^{3} +

9 x^{2} +

10 x -

12

ⓑ

(f \cdot g) (1) = 14

5.3-285 §5.3 source ↗

Multiply Polynomial Functions

Multiply the following:

$\begin{matrix} (x + 2) (x - 2) \\ (y + 7) (y - 7) \\ (w + 5) (w - 5) \end{matrix}$

Explain the pattern that you see in your answers.

Answer

Answers will vary.

5.3-287 §5.3 source ↗

Multiply Polynomial Functions

Why does ${(a + b)}^{2}$ result in a trinomial, but $(a - b) (a + b)$ result in a binomial?

Answer

Answers will vary.

Be Prepared 5.10 §5.4 source ↗

Add:

\frac{3}{d} + \frac{x}{d} .

If you missed this problem, review Example 1.28.

Example 1.28 in the book ↗

Answer

\frac{3 + x}{d}

Be Prepared 5.11 §5.4 source ↗

Simplify:

\frac{30 x y^{3}}{5 x y} .

If you missed this problem, review Example 1.25.

Example 1.25 in the book ↗

Answer

6 y^{2}

Be Prepared 5.12 §5.4 source ↗

Combine like terms:

8 a^{2} +

12 a +

1 +

3 a^{2} -

5 a +

4 .

If you missed this problem, review Example 1.7.

Example 1.7 in the book ↗

Answer

11 a^{2} + 7 a + 5

Try It 5.71 §5.4 source ↗

Find the quotient:

−72 a^{7} b^{3} \div (8 a^{12} b^{4}) .

Answer

- \frac{9}{a^{5} b}

Try It 5.72 §5.4 source ↗

Find the quotient:

−63 c^{8} d^{3} \div (7 c^{12} d^{2}) .

Answer

\frac{−9 d}{c^{4}}

Try It 5.73 §5.4 source ↗

Find the quotient:

\frac{28 x^{5} y^{14}}{49 x^{9} y^{12}} .

Answer

\frac{4 y^{2}}{7 x^{4}}

Try It 5.74 §5.4 source ↗

Find the quotient:

\frac{30 m^{5} n^{11}}{48 m^{10} n^{14}} .

Answer

\frac{5}{8 m^{5} n^{3}}

Try It 5.75 §5.4 source ↗

Find the quotient:

(32 a^{2} b - 16 a b^{2}) \div (−8 a b) .

Answer

−4 a + 2 b

Try It 5.76 §5.4 source ↗

Find the quotient:

(−48 a^{8} b^{4} - 36 a^{6} b^{5}) \div (−6 a^{3} b^{3}) .

Answer

8 a^{5} b + 6 a^{3} b^{2}

Try It 5.77 §5.4 source ↗

Find the quotient:

(y^{2} + 10 y + 21) \div (y + 3) .

Answer

y + 7

Try It 5.78 §5.4 source ↗

Find the quotient:

(m^{2} + 9 m + 20) \div (m + 4) .

Answer

m + 5

Try It 5.79 §5.4 source ↗

Find the quotient:

(x^{4} - 7 x^{2} + 7 x + 6) \div (x + 3) .

Answer

x^{3} -

3 x^{2} +

2 x +

1 +

\frac{3}{x + 3}

Try It 5.80 §5.4 source ↗

Find the quotient:

(x^{4} - 11 x^{2} - 7 x - 6) \div (x + 3) .

Answer

x^{3} -

3 x^{2} -

2 x -

1 -

\frac{3}{x + 3}

Try It 5.81 §5.4 source ↗

Find the quotient:

(x^{3} - 64) \div (x - 4) .

Answer

x^{2} + 4 x + 16

Try It 5.82 §5.4 source ↗

Find the quotient:

(125 x^{3} - 8) \div (5 x - 2) .

Answer

25 x^{2} + 10 x + 4

Try It 5.83 §5.4 source ↗

Use synthetic division to find the quotient and remainder when

3 x^{3} +

10 x^{2} +

6 x -

2

is divided by

x + 2 .

Answer

3 x^{2} + 4 x - 2; 2

Try It 5.84 §5.4 source ↗

Use synthetic division to find the quotient and remainder when

4 x^{3} +

5 x^{2} -

5 x +

3

is divided by

x + 2 .

Answer

4 x^{2} - 3 x + 1; 1

Try It 5.85 §5.4 source ↗

Use synthetic division to find the quotient and remainder when

x^{4} -

16 x^{2} +

5 x +

20

is divided by

x + 4 .

Answer

x^{3} - 4 x^{2} + 5; 0

Try It 5.86 §5.4 source ↗

Use synthetic division to find the quotient and remainder when

x^{4} - 9 x^{2} + 2 x + 6

is divided by

x + 3 .

Answer

x^{3} - 3 x^{2} + 2; 0

Try It 5.87 §5.4 source ↗

For functions

f (x) =

x^{2} -

5 x -

24

and

g (x) = x + 3,

find ⓐ

(\frac{f}{g}) (x)

ⓑ

(\frac{f}{g}) (−3) .

Answer

ⓐ

(\frac{f}{g}) (x) = x - 8

ⓑ

(\frac{f}{g}) (−3) =

−11

Try It 5.88 §5.4 source ↗

For functions

f (x) =

x^{2} -

5 x -

36

and

g (x) = x + 4,

find ⓐ

(\frac{f}{g}) (x)

ⓑ

(\frac{f}{g}) (−5) .

Answer

ⓐ

(\frac{f}{g}) (x) = x - 9

ⓑ undefined

Try It 5.89 §5.4 source ↗

Use the Remainder Theorem to find the remainder when

f (x) =

x^{3} +

4 x +

15

is divided by

x + 2 .

Answer

−1

Try It 5.90 §5.4 source ↗

Use the Remainder Theorem to find the remainder when

f (x) =

x^{3} -

7 x +

12

is divided by

x + 3 .

Answer

6

Try It 5.91 §5.4 source ↗

Use the Factor Theorem to determine if

x - 5

is a factor of

f (x) =

x^{3} -

125 .

Answer

yes

Try It 5.92 §5.4 source ↗

Use the Factor Theorem to determine if

x - 6

is a factor of

f (x) =

x^{3} -

216 .

Answer

yes

5.4-289 §5.4 source ↗

Divide Monomials

In the following exercises, divide the monomials.

$20 m^{8} n^{4} \div (30 m^{5} n^{9})$

Answer

\frac{2 m^{3}}{3 n^{5}}

5.4-291 §5.4 source ↗

Divide Monomials

In the following exercises, divide the monomials.

$\frac{45 x^{5} y^{9}}{−60 x^{8} y^{6}}$

Answer

\frac{−3 y^{3}}{4 x^{3}}

5.4-293 §5.4 source ↗

Divide Monomials

In the following exercises, divide the monomials.

$\frac{(−18 p^{4} q^{7}) (−6 p^{3} q^{8})}{−36 p^{12} q^{10}}$

Answer

\frac{−3 q^{5}}{p^{5}}

5.4-295 §5.4 source ↗

Divide Monomials

In the following exercises, divide the monomials.

$\frac{(4 u^{2} v^{5}) (15 u^{3} v)}{(12 u^{3} v) (u^{4} v)}$

Answer

\frac{5 v^{4}}{u^{2}}

5.4-297 §5.4 source ↗

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial.

$(8 x^{3} + 6 x^{2}) \div 2 x$

Answer

4 x^{2} + 3 x

5.4-299 §5.4 source ↗

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial.

$(48 y^{4} - 24 y^{3}) \div (−8 y^{2})$

Answer

−6 y^{2} + 3 y

5.4-301 §5.4 source ↗

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial.

$\frac{72 r^{5} s^{2} + 132 r^{4} s^{3} - 96 r^{3} s^{5}}{12 r^{2} s^{2}}$

Answer

6 r^{3} +

11 r^{2} s -

8 r s^{3}

5.4-303 §5.4 source ↗

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial.

$\frac{20 y^{2} + 12 y - 1}{−4 y}$

Answer

−5 y - 3 + \frac{1}{4 y}

5.4-305 §5.4 source ↗

Divide Polynomials using Long Division

In the following exercises, divide each polynomial by the binomial.

$(a^{2} - 2 a - 35) \div (a + 5)$

Answer

a - 7

5.4-307 §5.4 source ↗

Divide Polynomials using Long Division

In the following exercises, divide each polynomial by the binomial.

$(4 x^{2} - 17 x - 15) \div (x - 5)$

Answer

4 x + 3

5.4-309 §5.4 source ↗

Divide Polynomials using Long Division

In the following exercises, divide each polynomial by the binomial.

$(p^{2} + 11 p + 16) \div (p + 8)$

Answer

p + 3 - \frac{8}{p + 8}

5.4-311 §5.4 source ↗

Divide Polynomials using Long Division

In the following exercises, divide each polynomial by the binomial.

$(2 n^{3} - 10 n + 28) \div (n + 3)$

Answer

2 n^{2} -

6 n +

8 +

\frac{4}{n + 3}

5.4-313 §5.4 source ↗

Divide Polynomials using Long Division

In the following exercises, divide each polynomial by the binomial.

$(m^{3} + 1000) \div (m + 10)$

Answer

m^{2} - 10 m + 100

5.4-315 §5.4 source ↗

Divide Polynomials using Long Division

In the following exercises, divide each polynomial by the binomial.

$(125 y^{3} - 64) \div (5 y - 4)$

Answer

25 y^{2} + 20 y + 16

5.4-317 §5.4 source ↗

Divide Polynomials using Synthetic Division

In the following exercises, use synthetic Division to find the quotient and remainder.

$x^{3} -$ $3 x^{2} -$ $4 x +$ $12$ is divided by $x + 2$

Answer

x^{2} - 5 x + 6; 0

5.4-319 §5.4 source ↗

Divide Polynomials using Synthetic Division

In the following exercises, use synthetic Division to find the quotient and remainder.

$2 x^{3} -$ $11 x^{2} +$ $16 x -$ $12$ is divided by $x - 4$

Answer

2 x^{2} - 3 x + 4; 4

5.4-321 §5.4 source ↗

Divide Polynomials using Synthetic Division

In the following exercises, use synthetic Division to find the quotient and remainder.

$x^{4} + x^{2} + 6 x - 10$ is divided by $x + 2$

Answer

x^{3} -

2 x^{2} +

5 x -

4; − 2

5.4-323 §5.4 source ↗

Divide Polynomials using Synthetic Division

In the following exercises, use synthetic Division to find the quotient and remainder.

$3 x^{4} -$ $11 x^{3} +$ $2 x^{2} +$ $10 x +$ $6$ is divided by $x - 3$

Answer

3 x^{3} -

2 x^{2} -

4 x -

2; 0

5.4-325 §5.4 source ↗

Divide Polynomial Functions

In the following exercises, divide.

For functions $f (x) =$ $x^{2} -$ $15 x +$ $54$ and $g (x) = x - 9,$ find ⓐ $(\frac{f}{g}) (x)$ ⓑ $(\frac{f}{g}) (−5)$

Answer

ⓐ

(\frac{f}{g}) (x) = x - 6

ⓑ

(\frac{f}{g}) (−5) =

−11

5.4-327 §5.4 source ↗

Divide Polynomial Functions

In the following exercises, divide.

For functions $f (x) =$ $x^{3} +$ $2 x^{2} -$ $19 x +$ $12$ and $g (x) = x - 3,$ find ⓐ $(\frac{f}{g}) (x)$ ⓑ $(\frac{f}{g}) (0)$

Answer

ⓐ

(\frac{f}{g}) (x) =

x^{2} +

5 x -

4

ⓑ

(\frac{f}{g}) (0) = −4

5.4-329 §5.4 source ↗

Divide Polynomial Functions

In the following exercises, divide.

For functions $f (x) =$ $x^{2} +$ $2 x -$ $3$ and $g (x) = x + 3$ find ⓐ $(\frac{f}{g}) (x)$ ⓑ $(\frac{f}{g}) (3)$

Answer

ⓐ

\frac{f}{g} x =

x - 1;

x \neq - 3

ⓑ

\frac{f}{g} 3 = 2

5.4-331 §5.4 source ↗

Use the Remainder and Factor Theorem

In the following exercises, use the Remainder Theorem to find the remainder.

$f (x) =$ $x^{3} -$ $4 x -$ $9$ is divided by $x + 2$

Answer

−9

5.4-333 §5.4 source ↗

Use the Remainder and Factor Theorem

In the following exercises, use the Remainder Theorem to find the remainder.

$f (x) =$ $7 x^{2} -$ $5 x -$ $8$ divided by $x - 1$

Answer

−6

5.4-335 §5.4 source ↗

In the following exercises, use the Factor Theorem to determine if $x - c$ is a factor of the polynomial function.

Determine whether $x + 4$ a factor of $x^{3} + x^{2} - 14 x + 8$

Answer

5.4-337 §5.4 source ↗

In the following exercises, use the Factor Theorem to determine if $x - c$ is a factor of the polynomial function.

Determine whether $x - 3$ a factor of $x^{3} -$ $7 x^{2} +$ $11 x +$ $3$

Answer

yes

5.4-339 §5.4 source ↗

In the following exercises, use the Factor Theorem to determine if $x - c$ is a factor of the polynomial function.

Divide $\frac{10 x^{2} + x - 12}{2 x}$ and explain with words how you get each term of the quotient.

Answer

answer will vary

5.4-341 §5.4 source ↗

In the following exercises, use the Factor Theorem to determine if $x - c$ is a factor of the polynomial function.

In your own words, write the steps for synthetic division for $x^{2} + 5 x + 6$ divided by $x - 2 .$

Answer

Answers will vary.

5.R-343 §5.R source ↗

Types of Polynomials

In the following exercises, determine the type of polynomial.

$5 m + 9$

Answer

binomial

5.R-345 §5.R source ↗

Types of Polynomials

In the following exercises, determine the type of polynomial.

$y^{2} + 6 y^{3} + 9 y^{4}$

Answer

trinomial

5.R-347 §5.R source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$−8 y^{3} - 5 y^{3}$

Answer

−13 y^{3}

5.R-349 §5.R source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(8 m^{2} + 12 m - 5) -$ $(2 m^{2} - 7 m - 1)$

Answer

6 m^{2} + 19 m - 4

5.R-351 §5.R source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(5 u^{2} + 8 u) -$ $(4 u - 7)$

Answer

5 u^{2} + 4 u + 7

5.R-353 §5.R source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Find the difference of $x^{2} + 6 x + 8$ and $x^{2} - 8 x + 15 .$

Answer

14 x - 7

5.R-355 §5.R source ↗

In the following exercises, simplify.

$18 a - 7 b - 21 a$

Answer

−7 b - 3 a

5.R-357 §5.R source ↗

In the following exercises, simplify.

$(6 a^{2} + 7) +$ $(2 a^{2} - 5 a - 9)$

Answer

8 a^{2} - 5 a - 2

5.R-359 §5.R source ↗

In the following exercises, simplify.

$(7 m^{2} - 2 m - 5) -$ $(4 m^{2} + m - 8)$

Answer

3 m^{2} - 3 m + 3

5.R-361 §5.R source ↗

In the following exercises, simplify.

Subtract $(8 y^{2} - y + 9)$ from $(11 y^{2} - 9 y - 5)$

Answer

3 y^{2} - 8 y - 14

5.R-363 §5.R source ↗

In the following exercises, simplify.

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

Answer

x^{3} +

2 x^{2} y -

5 x y^{2} +

y^{3}

5.R-365 §5.R source ↗

Evaluate a Polynomial Function for a Given Value of the Variable

In the following exercises, find the function values for each polynomial function.

Answer

ⓐ 165 ⓑ 39 ⓒ 5

5.R-367 §5.R source ↗

Evaluate a Polynomial Function for a Given Value of the Variable

In the following exercises, find the function values for each polynomial function.

A pair of glasses is dropped off a bridge 640 feet above a river. The polynomial function $h (t) =$ $−16 t^{2} +$ $640$ gives the height of the glasses t seconds after they were dropped. Find the height of the glasses when $t = 6 .$

Answer

The height is

64 feet

5.R-369 §5.R source ↗

Add and Subtract Polynomial Functions

$f (x) =$ $2 x^{2} -$ $4 x -$ $7$ and $g (x) =$ $2 x^{2} -$ $x +$ $5$

Answer

ⓐ

(f + g) (x) =

4 x^{2} -

5 x -

2

ⓑ

(f + g) (3) = 19

ⓒ

(f - g) (x) =

−3 x -

12

ⓓ

(f - g) (−2) =

−6

5.R-371 §5.R source ↗

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

$p^{3} \cdot p^{10}$

Answer

p^{13}

5.R-373 §5.R source ↗

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

$a \cdot a^{2} \cdot a^{3}$

Answer

a^{6}

5.R-375 §5.R source ↗

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

$y^{a} \cdot y^{b}$

Answer

y^{a + b}

5.R-377 §5.R source ↗

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

$\frac{a^{6}}{a}$

Answer

a^{5}

5.R-379 §5.R source ↗

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

$\frac{1}{x^{5}}$

Answer

\frac{1}{x^{5}}

5.R-381 §5.R source ↗

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

$y^{0}$

Answer

5.R-383 §5.R source ↗

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

$12 a^{0} - 15 b^{0}$

Answer

−3

5.R-385 §5.R source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

${(−10)}^{−3}$

Answer

- \frac{1}{1000}

5.R-387 §5.R source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

${(8 n)}^{−1}$

Answer

\frac{1}{8 n}

5.R-389 §5.R source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

$10^{−3}$

Answer

\frac{1}{1000}

5.R-391 §5.R source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

$\frac{1}{6^{−2}}$

Answer

5.R-393 §5.R source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

${(- \frac{1}{5})}^{−3}$

Answer

−125

5.R-395 §5.R source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

${(−5)}^{−3}$

Answer

- \frac{1}{125}

5.R-397 §5.R source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

${(- \frac{3}{x})}^{−3}$

Answer

\frac{{- x}^{3}}{27}

5.R-399 §5.R source ↗

In the following exercises, simplify each expression using the Product Property.

${(3^{2})}^{5}$

Answer

3^{10}

5.R-401 §5.R source ↗

In the following exercises, simplify each expression using the Product Property.

$x^{−3} \cdot x^{9}$

Answer

x^{6}

5.R-403 §5.R source ↗

In the following exercises, simplify each expression using the Product Property.

$(u v^{−3}) (u^{−4} v^{−2})$

Answer

\frac{1}{u^{3} v^{5}}

5.R-405 §5.R source ↗

In the following exercises, simplify each expression using the Product Property.

$p^{5} \cdot p^{−2} \cdot p^{−4}$

Answer

\frac{1}{p}

5.R-407 §5.R source ↗

In the following exercises, simplify each expression using the Power Property.

$\frac{q^{4}}{q^{20}}$

Answer

\frac{1}{q^{16}}

5.R-409 §5.R source ↗

In the following exercises, simplify each expression using the Power Property.

$\frac{n^{−3}}{n^{−5}}$

Answer

n^{2}

5.R-411 §5.R source ↗

In the following exercises, simplify each expression using the Product to a Power Property.

${(−4 p q)}^{0}$

Answer

5.R-413 §5.R source ↗

In the following exercises, simplify each expression using the Product to a Power Property.

${(3 y^{−4})}^{2}$

Answer

\frac{9}{y^{8}}

5.R-415 §5.R source ↗

In the following exercises, simplify each expression using the Quotient to a Power Property.

${(\frac{3 x y^{2}}{z})}^{4}$

Answer

\frac{81 x^{4} y^{8}}{z^{4}}

5.R-417 §5.R source ↗

In the following exercises, simplify each expression by applying several properties.

${(x^{2} y)}^{2} {(3 x y^{5})}^{3}$

Answer

27 x^{7} y^{17}

5.R-419 §5.R source ↗

In the following exercises, simplify each expression by applying several properties.

${(\frac{3 x y^{3}}{4 x^{4} y^{−2}})}^{2} {(\frac{6 x y^{4}}{8 x^{3} y^{−2}})}^{−1}$

Answer

\frac{3 y^{4}}{4 x^{4}}

5.R-421 §5.R source ↗

In the following exercises, write each number in scientific notation.

5,300,000

Answer

5.3 \times 10^{6}

5.R-423 §5.R source ↗

In the following exercises, convert each number to decimal form.

$2.9 \times 10^{4}$

Answer

29,000

5.R-425 §5.R source ↗

In the following exercises, convert each number to decimal form.

$9.413 \times 10^{−5}$

Answer

0.00009413

5.R-427 §5.R source ↗

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

$(1.5 \times 10^{−3}) (4.8 \times 10^{−1})$

Answer

0.00072

5.R-429 §5.R source ↗

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

$\frac{9 \times 10^{−3}}{1 \times 10^{−6}}$

Answer

9,000

5.R-431 §5.R source ↗

Multiply Monomials

In the following exercises, multiply the monomials.

$(\frac{1}{3} c^{2}) (30 c^{8})$

Answer

10 c^{10}

5.R-433 §5.R source ↗

Multiply Monomials

In the following exercises, multiply the monomials.

$(\frac{2}{3} m^{3} n^{6}) (\frac{1}{6} m^{4} n^{4})$

Answer

\frac{m^{7} n^{10}}{9}

5.R-435 §5.R source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$a^{2} (a^{2} - 9 a - 36)$

Answer

a^{4} - 9 a^{3} - 36 a^{2}

5.R-437 §5.R source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$(4 n - 5) (2 n^{3})$

Answer

8 n^{4} - 10 n^{3}

5.R-439 §5.R source ↗

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using:

$(y - 4) (y + 12)$

Answer

y^{2} + 8 y - 48

5.R-441 §5.R source ↗

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using:

$(6 p - 11) (3 p - 10)$

Answer

18 p^{2} -

93 p +

110

5.R-443 §5.R source ↗

In the following exercises, multiply the binomials. Use any method.

$(k + 6) (k - 9)$

Answer

k^{2} - 3 k - 54

5.R-445 §5.R source ↗

In the following exercises, multiply the binomials. Use any method.

$(2 y - 9) (5 y - 7)$

Answer

10 y^{2} - 59 y + 63

5.R-447 §5.R source ↗

In the following exercises, multiply the binomials. Use any method.

$(x - 8) (x + 9)$

Answer

x^{2} + x - 72

5.R-449 §5.R source ↗

In the following exercises, multiply the binomials. Use any method.

$(10 a - 1) (3 a - 3)$

Answer

30 a^{2} - 33 a + 3

5.R-451 §5.R source ↗

Multiply a Polynomial by a Polynomial

In the following exercises, multiply using ⓐ the Distributive Property ⓑ the Vertical Method.

$(5 b - 2) (3 b^{2} + b - 9)$

Answer

15 b^{3} -

b^{2} -

47 b +

18

5.R-453 §5.R source ↗

In the following exercises, multiply. Use either method.

$(4 y - 1) (6 y^{2} - 12 y + 5)$

Answer

24 y^{3} -

54 y^{2} +

32 y -

5

5.R-455 §5.R source ↗

Multiply Special Products

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(x + \frac{3}{4})}^{2}$

Answer

x^{2} + \frac{3}{2} x + \frac{9}{16}

5.R-457 §5.R source ↗

Multiply Special Products

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(5 p + 7 q)}^{2}$

Answer

25 p^{2} +

70 p q +

49 q^{2}

5.R-459 §5.R source ↗

In the following exercises, multiply each pair of conjugates using the Product of Conjugates.

$(6 x + y) (6 x - y)$

Answer

36 x^{2} - y^{2}

5.R-461 §5.R source ↗

In the following exercises, multiply each pair of conjugates using the Product of Conjugates.

$(12 x^{3} - 7 y^{2}) (12 x^{3} + 7 y^{2})$

Answer

144 x^{6} - 49 y^{4}

5.R-463 §5.R source ↗

Divide Monomials

In the following exercises, divide the monomials.

$72 p^{12} \div 8 p^{3}$

Answer

9 p^{9}

5.R-465 §5.R source ↗

Divide Monomials

In the following exercises, divide the monomials.

$\frac{45 y^{6}}{−15 y^{10}}$

Answer

- \frac{3}{y^{4}}

5.R-467 §5.R source ↗

Divide Monomials

In the following exercises, divide the monomials.

$\frac{28 a^{9} b}{7 a^{4} b^{3}}$

Answer

\frac{4 a^{5}}{b^{2}}

5.R-469 §5.R source ↗

Divide Monomials

In the following exercises, divide the monomials.

$\frac{(5 m^{9} n^{3}) (8 m^{3} n^{2})}{(10 m n^{4}) (m^{2} n^{5})}$

Answer

\frac{4 m^{9}}{n^{4}}

5.R-471 §5.R source ↗

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial

$(54 y^{4} - 24 y^{3}) \div (−6 y^{2})$

Answer

−9 y^{2} + 4 y

5.R-473 §5.R source ↗

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial

$\frac{12 x^{2} + 4 x - 3}{−4 x}$

Answer

−3 x - 1 + \frac{3}{4 x}

5.R-475 §5.R source ↗

Divide Polynomials using Long Division

In the following exercises, divide each polynomial by the binomial.

$(y^{2} + 2 y + 18) \div (y + 5)$

Answer

y - 3 + \frac{33}{y + 5}

5.R-477 §5.R source ↗

Divide Polynomials using Long Division

In the following exercises, divide each polynomial by the binomial.

$(a^{3} - 1) \div (a + 1)$

Answer

a^{2} - a + 1 - \frac{2}{a + 1}

5.R-479 §5.R source ↗

Divide Polynomials using Synthetic Division

In the following exercises, use synthetic Division to find the quotient and remainder.

$2 x^{3} -$ $11 x^{2} +$ $11 x +$ $12$ is divided by $x - 3$

Answer

2 x^{2} - 5 x - 4; 0

5.R-481 §5.R source ↗

Divide Polynomial Functions

In the following exercises, divide.

For functions $f (x) =$ $x^{2} -$ $15 x +$ $54$ and $g (x) = x - 9,$ find ⓐ $(\frac{f}{g}) (x)$
ⓑ $(\frac{f}{g}) (−2)$

Answer

ⓐ

(\frac{f}{g}) (x) = x - 6

ⓑ

(\frac{f}{g}) (−2) = −8

5.R-483 §5.R source ↗

Use the Remainder and Factor Theorem

In the following exercises, use the Remainder Theorem to find the remainder.

$f (x) =$ $x^{3} -$ $4 x -$ $9$ is divided by $x + 2$

Answer

−9

5.R-485 §5.R source ↗

In the following exercises, use the Factor Theorem to determine if $x - c$ is a factor of the polynomial function.

Determine whether $x - 2$ is a factor of $x^{3} - 7 x^{2} + 7 x - 6$ .

Answer

5.T-487 §5.T source ↗

For the polynomial $8 y^{4} - 3 y^{2} + 1$

ⓐ Is it a monomial, binomial, or trinomial? ⓑ What is its degree?

Answer

ⓐ trinomial ⓑ 4

5.T-489 §5.T source ↗

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

(4 x^{2} - 6)

Answer

6 x^{2} - 3 x + 11

5.T-491 §5.T source ↗

x^{−3} x^{4}

Answer

x

5.T-493 §5.T source ↗

{(47 a^{18} b^{23} c^{5})}^{0}

Answer

1

5.T-495 §5.T source ↗

{(2 y)}^{−3}

Answer

\frac{1}{8 y^{3}}

5.T-497 §5.T source ↗

\frac{x^{4}}{x^{−5}}

Answer

x^{9}

5.T-499 §5.T source ↗

\frac{24 r^{3} s}{6 r^{2} s^{7}}

Answer

\frac{4 r}{s^{6}}

5.T-501 §5.T source ↗

(8 x y^{3}) (−6 x^{4} y^{6})

Answer

−48 x^{5} y^{9}

5.T-503 §5.T source ↗

(m + 3) (7 m - 2)

Answer

7 m^{2} + 19 m - 6

5.T-505 §5.T source ↗

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

Answer

16 x^{2} - 24 x + 9

5.T-507 §5.T source ↗

(15 x y^{3} - 35 x^{2} y) \div 5 x y

Answer

3 y^{2} - 7 x

5.T-509 §5.T source ↗

Use the Factor Theorem to determine if

x + 3

a factor of

x^{3} +

8 x^{2} +

21 x +

18 .

Answer

yes

5.T-511 §5.T source ↗

In the following exercises, simplify and write your answer in exponential notation.

$(2.4 \times 10^{8}) (2 \times 10^{−5})$

Answer

4.8 \times 10^{3}

5.T-513 §5.T source ↗

In the following exercises, simplify and write your answer in exponential notation.

Answer

ⓐ

36

ⓑ

21

ⓒ

- 9

5.T-515 §5.T source ↗

In the following exercises, simplify and write your answer in exponential notation.

For functions
$f (x) =$ $3 x^{2} -$ $23 x -$ $36$ and
$g (x) = x - 9,$ find
ⓐ $(\frac{f}{g}) (x)$ ⓑ $(\frac{f}{g}) (3)$

Answer

ⓐ

(\frac{f}{g}) (x) =

3 x +

4

ⓑ

(\frac{f}{g}) (3) = 13

Loading chapter 6…

Problems and answers are adapted from Intermediate Algebra 2e by Lynn Marecek and Andrea Honeycutt Mathis, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.