Skip to main content
$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# Chapter 5 Review Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

## Chapter Review Exercises

### Add and Subtract Polynomials

Determine the Degree of Polynomials

In the following exercises, determine the type of polynomial.

1.  $$16x^2−40x−25$$

2.  $$5m+9$$

Answer

binomial

3.  $$−15$$

4.  $$y^2+6y^3+9y^4$$

Answer

other polynomial

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

5.  $$4p+11p$$

6.  $$−8y^3−5y^3$$

Answer

$$−13y^3$$

7.  $$(4a^2+9a−11)+(6a^2−5a+10)$$

8.  $$(8m^2+12m−5)−(2m^2−7m−1)$$

Answer

$$6m^2+19m−4$$

9.  $$(y^2−3y+12)+(5y^2−9)$$

10.  $$(5u^2+8u)−(4u−7)$$

Answer

$$5u^2+4u+7$$

11. Find the sum of $$8q^3−27$$ and $$q^2+6q−2$$.

12. Find the difference of $$x^2+6x+8$$ and $$x^2−8x+15$$.

Answer

$$2x^2−2x+23$$

In the following exercises, simplify.

13.  $$17mn^2−(−9mn^2)+3mn^2$$

14.  $$18a−7b−21a$$

Answer

$$−7b−3a$$

15.  $$2pq^2−5p−3q^2$$

16.  $$(6a^2+7)+(2a^2−5a−9)$$

Answer

$$8a^2−5a−2$$

17.  $$(3p^2−4p−9)+(5p^2+14)$$

18.  $$(7m^2−2m−5)−(4m^2+m−8)$$

Answer

$$−3m+3$$

19.  $$(7b^2−4b+3)−(8b^2−5b−7)$$

20. Subtract $$(8y^2−y+9)$$ from $$(11y^2−9y−5)$$

Answer

$$3y^2−8y−14$$

21. Find the difference of $$(z^2−4z−12)$$ and $$(3z^2+2z−11)$$

22.  $$(x^3−x^2y)−(4xy^2−y^3)+(3x^2y−xy^2)$$

Answer

$$x^3+2x^2y−4xy^2$$

23.  $$(x^3−2x^2y)−(xy^2−3y^3)−(x^2y−4xy^2)$$

Evaluate a Polynomial Function for a Given Value of the Variable

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

24. For the function $$f(x)=7x^2−3x+5$$ find:
a. $$f(5)$$   b. $$f(−2)$$   c. $$f(0)$$

Answer

a. 165   b. 39   c. 5

25. For the function $$g(x)=15−16x^2$$, find:
a. $$g(−1)$$   b. $$g(0)$$   c. $$g(2)$$

26. A pair of glasses is dropped off a bridge 640 feet above a river. The polynomial function $$h(t)=−16t^2+640$$ 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.

27. A manufacturer of the latest soccer shoes has found that the revenue received from selling the shoes at a cost of $$p$$ dollars each is given by the polynomial $$R(p)=−5p^2+360p$$. Find the revenue received when $$p=110$$ dollars.

Add and Subtract Polynomial Functions

In the following exercises, find a. $$(f + g)(x)$$ b. $$(f + g)(3)$$ c. $$(f − g)(x$$ d. $$(f − g)(−2)$$

28.  $$f(x)=2x^2−4x−7$$ and $$g(x)=2x^2−x+5$$

Answer

a. $$(f+g)(x)=4x^2−5x−2$$
b. $$(f+g)(3)=19$$
c. $$(f−g)(x)=−3x−12$$
d. $$(f−g)(−2)=−6$$

29.  $$f(x)=4x^3−3x^2+x−1$$ and $$g(x)=8x^3−1$$

### Properties of Exponents and Scientific Notation

Simplify Expressions Using the Properties for Exponents

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

30.  $$p^3·p^{10}$$

Answer

$$p^{13}$$

31.  $$2·2^6$$

32.  $$a·a^2·a^3$$

Answer

$$a^6$$

33.  $$x·x^8$$

34.  $$y^a·y^b$$

Answer

$$y^{a+b}$$

35.  $$\dfrac{2^8}{2^2}$$

36.  $$\dfrac{a^6}{a}$$

Answer

$$a^5$$

37.  $$\dfrac{n^3}{n^{12}}$$

38.  $$\dfrac{1}{x^5}$$

Answer

$$\dfrac{1}{x^4}$$

39.  $$3^0$$

40.  $$y^0$$

Answer

$$1$$

41.  $$(14t)^0$$

42.  $$12a^0−15b^0$$

Answer

$$−3$$

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

43.  $$6^{−2}$$

44.  $$(−10)^{−3}$$

Answer

$$−\dfrac{1}{1000}$$

45.  $$5·2^{−4}$$

46.  $$(8n)^{−1}$$

Answer

$$\dfrac{1}{8n}$$

47.  $$y^{−5}$$

48.  $$10^{−3}$$

Answer

$$\dfrac{1}{1000}$$

49.  $$\dfrac{1}{a^{−4}}$$

50.  $$\dfrac{1}{6^{−2}}$$

Answer

$$36$$

51.  $$−5^{−3}$$

52.  $$\left(−\dfrac{1}{5}\right)^{−3}$$

Answer

$$−\dfrac{1}{25}$$

53.  $$−(12)^{−3}$$

54.  $$(−5)^{−3}$$

Answer

$$−\dfrac{1}{125}$$

55.  $$\left(\dfrac{5}{9}\right)^{−2}$$

56.  $$\left(−\dfrac{3}{x}\right)^{−3}$$

Answer

$$\dfrac{x^3}{27}$$

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

57.  $$(y^4)^3$$

58.  $$(3^2)^5$$

Answer

$$3^{10}$$

59.  $$(a^{10})^y$$

60.  $$x^{−3}·x^9$$

Answer

$$x^5$$

61.  $$r^{−5}·r^{−4}$$

62.  $$(uv^{−3})(u^{−4}v^{−2})$$

Answer

$$\dfrac{1}{u^3v^5}$$

63.  $$(m^5)^{−1}$$

64.  $$p^5·p^{−2}·p^{−4}$$

Answer

$$\dfrac{1}{m^5}$$

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

65.  $$(k−2)^{−3}$$

66.  $$\dfrac{q^4}{q^{20}}$$

Answer

$$\dfrac{1}{q^{16}}$$

67.  $$\dfrac{b^8}{b^{−2}}$$

68.  $$\dfrac{n^{−3}}{n^{−5}}$$

Answer

$$n^2$$

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

69.  $$(−5ab)^3$$

70.  $$(−4pq)^0$$

Answer

$$1$$

71.  $$(−6x^3)^{−2}$$

72.  $$(3y^{−4})^2$$

Answer

$$\dfrac{9}{y^8}$$

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

73.  $$\left(\dfrac{3}{5x}\right)^{−2}$$

74.  $$\left(\dfrac{3xy^2}{z}\right)^4$$

Answer

$$\dfrac{81x^4y^8}{z^4}$$

75.  $$(4p−3q^2)^2$$

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

76.  $$(x^2y)^2(3xy^5)^3$$

Answer

$$27x^7y^{17}$$

77.  $$(−3a^{−2})^4(2a^4)^2(−6a^2)^3$$

78.  $$\left(\dfrac{3xy^3}{4x^4y^{−2}}\right)^2\left(\dfrac{6xy^4}{8x^3y^{−2}}\right)^{−1}$$

Answer

$$\dfrac{3y^4}{4x^4}$$

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

79.  $$2.568$$

80.  $$5,300,000$$

Answer

$$5.3×10^6$$

81.  $$0.00814$$

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

82.  $$2.9×10^4$$

Answer

$$29,000$$

83.  $$3.75×10^{−1}$$

84.  $$9.413×10^{−5}$$

Answer

$$0.00009413$$

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

85.  $$(3×10^7)(2×10^{−4})$$

86.  $$(1.5×10^{−3})(4.8×10^{−1})$$

Answer

$$0.00072$$

87.  $$\dfrac{6×10^9}{2×10^{−1}}$$

88.  $$\dfrac{9×10^{−3}}{1×10^{−6}}$$

Answer

$$9,000$$

### Multiply Polynomials

Multiply Monomials

In the following exercises, multiply the monomials.

89.  $$(−6p^4)(9p)$$

90.  $$\left(\frac{1}{3}c^2\right)(30c^8)$$

Answer

$$10c^{10}$$

91.  $$(8x^2y^5)(7xy^6)$$

92.  $$\left(\frac{2}{3}m^3n^6\right)\left(\frac{1}{6}m^4n^4\right)$$

Answer

$$\dfrac{m^7n^{10}}{9}$$

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

93.  $$7(10−x)$$

94.  $$a^2(a^2−9a−36)$$

Answer

$$a^4−9a^3−36a^2$$

95.  $$−5y(125y^3−1)$$

96.  $$(4n−5)(2n^3)$$

Answer

$$8n^4−10n^3$$

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using:

a. the Distributive Property b. the FOIL method c. the Vertical Method.

97.  $$(a+5)(a+2)$$

98.  $$(y−4)(y+12)$$

Answer

$$y^2+8y−48$$

99.  $$(3x+1)(2x−7)$$

100.  $$(6p−11)(3p−10)$$

Answer

$$18p^2−93p+110$$

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

101.  $$(n+8)(n+1)$$

102.  $$(k+6)(k−9)$$

Answer

$$k^2−3k−54$$

103.  $$(5u−3)(u+8)$$

104.  $$(2y−9)(5y−7)$$

Answer

$$10y^2−59y+63$$

105.  $$(p+4)(p+7)$$

106.  $$(x−8)(x+9)$$

Answer

$$x^2+x−72$$

107.  $$(3c+1)(9c−4)$$

108.  $$(10a−1)(3a−3)$$

Answer

$$30a^2−33a+3$$

Multiply a Polynomial by a Polynomial

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

109.  $$(x+1)(x^2−3x−21)$$

110.  $$(5b−2)(3b^2+b−9)$$

Answer

$$15b^3−b^2−47b+18$$

In the following exercises, multiply. Use either method.

111.  $$(m+6)(m^2−7m−30)$$

112.  $$(4y−1)(6y^2−12y+5)$$

Answer

$$24y^2−54y^2+32y−5$$

Multiply Special Products

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

113.  $$(2x−y)^2$$

114.  $$(x+\dfrac{3}{4})^2$$

Answer

$$x^2+\dfrac{3}{2}x+\dfrac{9}{16}$$

115.  $$(8p^3−3)^2$$

116.  $$(5p+7q)^2$$

Answer

$$25p^2+70pq+49q^2$$

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

117.  $$(3y+5)(3y−5)$$

118.  $$(6x+y)(6x−y)$$

Answer

$$36x^2−y^2$$

119.  $$(a+\dfrac{2}3b)(a−\dfrac{2}{3}b)$$

120.  $$(12x^3−7y^2)(12x^3+7y^2)$$

Answer

$$144x^6−49y^4$$

121.  $$(13a^2−8b4)(13a^2+8b^4)$$

### Divide Monomials

Divide Monomials

In the following exercises, divide the monomials.

122.  $$72p^{12}÷8p^3$$

Answer

$$9p^9$$

123.  $$−26a^8÷(2a^2)$$

124.  $$\dfrac{45y^6}{−15y^{10}}$$

Answer

$$−3y^4$$

125.  $$\dfrac{−30x^8}{−36x^9}$$

126.  $$\dfrac{28a^9b}{7a^4b^3}$$

Answer

$$\dfrac{4a^5}{b^2}$$

127.  $$\dfrac{11u^6v^3}{55u^2v^8}$$

128.  $$\dfrac{(5m^9n^3)(8m^3n^2)}{(10mn^4)(m^2n^5)}$$

Answer

$$\dfrac{4m^9}{n^4}$$

129.  $$\dfrac{(42r^2s^4)(54rs^2)}{(6rs^3)(9s)}$$

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial

130.  $$(54y^4−24y^3)÷(−6y^2)$$

Answer

$$−9y^2+4y$$

131.  $$\dfrac{63x^3y^2−99x^2y^3−45x^4y^3}{9x^2y^2}$$

132.  $$\dfrac{12x^2+4x−3}{−4x}$$

Answer

$$−3x−1+\dfrac{3}{4x}$$

Divide Polynomials using Long Division

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

133.  $$(4x^2−21x−18)÷(x−6)$$

134.  $$(y^2+2y+18)÷(y+5)$$

Answer

$$y−3+\dfrac{33}{q+6}$$

135.  $$(n^3−2n^2−6n+27)÷(n+3)$$

136.  $$(a^3−1)÷(a+1)$$

Answer

$$a^2+a+1$$

Divide Polynomials using Synthetic Division

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

137.  $$x^3−3x^2−4x+12$$ is divided by $$x+2$$

138.  $$2x^3−11x^2+11x+12$$ is divided by $$x−3$$

Answer

$$2x^2−5x−4;\space0$$

139.  $$x^4+x^2+6x−10$$ is divided by $$x+2$$

Divide Polynomial Functions

In the following exercises, divide.

140. For functions $$f(x)=x^2−15x+45$$ and $$g(x)=x−9$$, find a. $$\left(\dfrac{f}{g}\right)(x)$$
b. $$\left(\dfrac{f}{g}\right)(−2)$$

Answer

a. $$\left(\dfrac{f}{g}\right)(x)=x−6$$
b. $$\left(\dfrac{f}{g}\right)(−2)=−8$$

141. For functions $$f(x)=x^3+x^2−7x+2$$ and $$g(x)=x−2$$, find a. $$\left(\dfrac{f}{g}\right)(x)$$
b. $$\left(\dfrac{f}{g}\right)(3)$$

Use the Remainder and Factor Theorem

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

142.  $$f(x)=x^3−4x−9$$ is divided by $$x+2$$

Answer

$$−9$$

143.  $$f(x)=2x^3−6x−24$$ divided by $$x−3$$

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

144. Determine whether $$x−2$$ is a factor of $$x^3−7x^2+7x−6$$

Answer

no

145. Determine whether $$x−3$$ is a factor of $$x^3−7x^2+11x+3$$

## Chapter Practice Test

1. For the polynomial $$8y^4−3y^2+1$$

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

Answer

a. trinomial b. 4

2.  $$(5a^2+2a−12)(9a^2+8a−4)$$

3.  $$(10x^2−3x+5)−(4x^2−6)$$

Answer

$$6x^2−3x+11$$

4.  $$\left(−\dfrac{3}{4}\right)^3$$

5.  $$x^{−3}x^4$$

Answer

$$x$$

6.  $$5^65^8$$

7.  $$(47a^{18}b^{23}c^5)^0$$

Answer

$$1$$

8.  $$4^{−1}$$

9.  $$(2y)^{−3}$$

Answer

$$\dfrac{1}{8y^3}$$

10.  $$p^{−3}·p^{−8}$$

11.  $$\dfrac{x^4}{x^{−5}}$$

Answer

$$x^9$$

12.  $$(3x^{−3})^2$$

13.  $$\dfrac{24r^3s}{6r^2s^7}$$

Answer

$$\dfrac{4r}{s^6}$$

14.  $$(x4y9x−3)2$$

15.  $$(8xy^3)(−6x^4y^6)$$

Answer

$$−48x^5y^9$$

16.  $$4u(u^2−9u+1)$$

17.  $$(m+3)(7m−2)$$

Answer

$$21m^2−19m−6$$

18.  $$(n−8)(n^2−4n+11)$$

19.  $$(4x−3)^2$$

Answer

$$16x^2−24x+9$$

20.  $$(5x+2y)(5x−2y)$$

21.  $$(15xy^3−35x^2y)÷5xy$$

Answer

$$3y^2−7x$$

22.  $$(3x^3−10x^2+7x+10)÷(3x+2)$$

23. Use the Factor Theorem to determine if $$x+3$$ a factor of $$x^3+8x^2+21x+18$$.

Answer

yes

24.  a. Convert 112,000 to scientific notation.
b. Convert $$5.25×10^{−4}$$ to decimal form.

In the following exercises, simplify and write your answer in decimal form.

25.  $$(2.4×10^8)(2×10^{−5})$$

Answer

$$4.4×10^3$$

26.  $$\dfrac{9×10^4}{3×10^{−1}}$$

27. For the function $$f(x)=6x^2−3x−9$$ find:
a. $$f(3)$$    b. $$f(−2)$$    c. $$f(0)$$

Answer

a. $$36$$    b. $$21$$    c. $$-9$$

28. For $$f(x)=2x^2−3x−5$$ and $$g(x)=3x^2−4x+1$$, find
a. $$(f+g)(x)$$    b. $$(f+g)(1)$$
c. $$(f−g)(x)$$    d. $$(f−g)(−2)$$

29. For functions $$f(x)=3x^2−23x−36$$ and $$g(x)=x−9$$, find
a. $$\left(\dfrac{f}{g}\right)(x)$$    b. $$\left(\dfrac{f}{g}\right)(3)$$

Answer

a. $$\left(\dfrac{f}{g}\right)(x)=3x+4$$
b. $$\left(\dfrac{f}{g}\right)(3)=13$$

30. A hiker drops a pebble from a bridge $$240$$ feet above a canyon. The function $$h(t)=−16t^2+240$$ gives the height of the pebble $$t$$ seconds after it was dropped. Find the height when $$t=3$$.