Algebra 2

Holliday, Luchin, Cuevas, Carter Marks, Day, Casey, Hayek

Chapter 6

Polynomial Functions - all with Video Answers

Educators

Section 1

Properties of Exponents

Problem 1

Simplify. Assume that no variable equals 0.
$$
\left(-3 x^{2} y^{3}\right)\left(5 x^{5} y^{6}\right)
$$

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Problem 2

Simplify. Assume that no variable equals 0.
$$
\frac{30 y^{4}}{-5 y^{2}}
$$

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Problem 3

Simplify. Assume that no variable equals 0.
$$
\frac{-2 a^{3} b^{6}}{18 a^{2} b^{2}}
$$

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Problem 4

Simplify. Assume that no variable equals 0.
$$
(2 b)^{4}
$$

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Problem 5

Simplify. Assume that no variable equals 0.
$$
\left(\frac{1}{w^{4} z^{2}}\right)^{3}
$$

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Problem 6

Simplify. Assume that no variable equals 0.
$$
\left(\frac{c d}{3}\right)^{-2}
$$

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Problem 7

Simplify. Assume that no variable equals 0.
$$
\left(n^{3}\right)^{3}\left(n^{-3}\right)^{3}
$$

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Problem 8

Simplify. Assume that no variable equals 0.
$$
\frac{81 p^{6} q^{5}}{\left(3 p^{2} q\right)^{2}}
$$

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Problem 9

Simplify. Assume that no variable equals 0.
$$
\left(\frac{-6 x^{6}}{3 x^{3}}\right)^{-2}
$$

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Problem 10

ASTRONOMY. Refer to Example 5 on page $315 .$ The average distance from Earth to the Moon is about $3.84 \times 10^{8}$ meters. How long would it take a radio signal traveling at the speed of light to cover that distance?

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Problem 11

Simplify. Assume that no variable equals 0.
$$
\left(\frac{1}{3} a^{8} b^{2}\right)\left(2 a^{2} b^{2}\right)
$$

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Problem 12

Simplify. Assume that no variable equals 0.
$$
\left(5 c d^{2}\right)\left(-c^{4} d\right)
$$

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Problem 13

Simplify. Assume that no variable equals 0.
$$
\left(7 x^{3} y^{-5}\right)\left(4 x y^{3}\right)
$$

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Problem 14

Simplify. Assume that no variable equals 0.
$$
\left(-3 b^{3} c\right)\left(7 b^{2} c^{2}\right)
$$

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Problem 15

Simplify. Assume that no variable equals 0.
$$
\frac{a^{2} n^{6}}{a n^{5}}
$$

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Problem 16

Simplify. Assume that no variable equals 0.
$$
\frac{-y^{5} z^{7}}{y^{2} z^{5}}
$$

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Problem 17

Simplify. Assume that no variable equals 0.
$$
\frac{-5 x^{3} y^{3} z^{4}}{20 x^{3} y^{7} z^{4}}
$$

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Problem 18

Simplify. Assume that no variable equals 0.
$$
\frac{3 a^{5} b^{3} c^{3}}{9 a^{3} b^{7} c}
$$

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Problem 19

Simplify. Assume that no variable equals 0.
$$
\left(n^{4}\right)^{4}
$$

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Problem 20

Simplify. Assume that no variable equals 0.
$$
\left(z^{2}\right)^{5}
$$

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Problem 21

Simplify. Assume that no variable equals 0.
$$
(2 x)^{4}
$$

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Problem 22

Simplify. Assume that no variable equals 0.
$$
(-2 c)^{3}
$$

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Problem 23

Simplify. Assume that no variable equals 0.
$$
\left(a^{3} b^{3}\right)(a b)^{-2}
$$

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Problem 24

Simplify. Assume that no variable equals 0.
$$
\left(-2 r^{2} s\right)^{3}\left(3 r s^{2}\right)
$$

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Problem 25

Simplify. Assume that no variable equals 0.
$$
\frac{2 c^{3} d\left(3 c^{2} d^{5}\right)}{30 c^{4} d^{2}}
$$

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Problem 26

Simplify. Assume that no variable equals 0.
$$
\frac{-12 m^{4} n^{8}\left(m^{3} n^{2}\right)}{36 m^{3} n}
$$

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Problem 27

BIOLOGY. Use the diagram at the right to write the diameter of a typical flu virus in scientific notation. Then estimate the area of a typical flu virus. (Hint: Treat the virus as a circle.)

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Problem 28

POPULATION. The population of Earth is about $6.445 \times 10^{9} .$ The land surface area of Earth is $1.483 \times 10^{8} \mathrm{km}^{2} .$ What is the population density for the land surface area of Earth?

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Problem 29

Simplify. Assume that no variable equals 0.
$$
2 x^{2}\left(6 y^{3}\right)\left(2 x^{2} y\right)
$$

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Problem 30

Simplify. Assume that no variable equals 0.
$$
3 a\left(5 a^{2} b\right)\left(6 a b^{3}\right)
$$

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Problem 31

Simplify. Assume that no variable equals 0.
$$
\frac{30 a^{-2} b^{-6}}{60 a^{-6} b^{-8}}
$$

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Problem 32

Simplify. Assume that no variable equals 0.
$$
\frac{12 x^{-3} y^{-2} z^{-8}}{30 x^{-6} y^{-4} z^{-1}}
$$

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Problem 33

Simplify. Assume that no variable equals 0.
$$
\left(\frac{x}{y^{-1}}\right)^{-2}
$$

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Problem 34

Simplify. Assume that no variable equals 0.
$$
\left(\frac{v}{w^{-2}}\right)^{-3}
$$

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Problem 35

Simplify. Assume that no variable equals 0.
$$
\left(\frac{8 a^{3} b^{2}}{16 a^{2} b^{3}}\right)^{4}
$$

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Problem 36

Simplify. Assume that no variable equals 0.
$$
\left(\frac{6 x^{2} y^{4}}{3 x^{4} y^{3}}\right)^{3}
$$

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Problem 37

Simplify. Assume that no variable equals 0.
$$
\left(\frac{4 x^{-3} y^{2}}{x y^{-5}}\right)^{-2}
$$

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Problem 38

If $2^{r+5}=2^{2 r-1},$ what is the value of $r ?$

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Problem 39

What value of $r$ makes $y^{28}=y^{3 r} \cdot y^{7}$ true?

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Problem 40

INCOME In $2003,$ the population of Texas was about $2.21 \times 10^{7}$ . The personal income for the state that year was about $6.43 \times 10^{11}$ dollars. What was the average personal income?

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Problem 41

RESEARCH Use the Internet or other source to find the masses of Earth and the Sun. About how many times as large as Earth is the Sun?

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Problem 42

OPEN ENDED Write an example that illustrates a property of powers. Then use multiplication or division to explain why it is true.

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Problem 43

FIND THE ERROR. Alejandra and Kyle both simplified $\frac{2 a^{2} b}{\left(-2 a^{3} b\right)^{-2}} .$ Who is correct? Explain your reasoning.
$$
\begin{array}{l}{\text { Alejandra }} \\ {\begin{aligned} \frac{2 a^{2} b}{\left(-2 a b^{3}\right)^{2}} &=\left(2 a^{2} b\right)\left(-2 a b^{3}\right)^{2} \\ &=\left(2 a^{2} b\right)(-2)^{2} a^{2}\left(b^{3}\right)^{2} \\ &=\left(2 a^{2} b\right) 2^{2} a^{2} b^{6} \\ &=8 a^{4} b^{7} \end{aligned}}\end{array}
$$
$$
\begin{aligned} & \text { Kyle } \\ \frac{2 a^{2} b}{\left(-2 a b^{3}\right)^{-2}} &=\frac{2 a^{2} b}{(-2)^{2} a\left(b^{3}\right)^{-2}} \\ &=\frac{2 a^{2} b}{4 a b^{-6}} \\ &=\frac{2 a^{2} b b^{6}}{4 a} \\ &=\frac{a b^{7}}{2} \end{aligned}
$$

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Problem 44

REASONING. Determine whether $x^{y} \cdot x^{z}=x^{y z}$ is sometimes, always, or never true. Explain your reasoning.

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Problem 45

CHALLENGE Determine which is greater, $100^{10}$ or $10^{100} .$ Explain.

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Problem 46

Writing in Math. Use the information on page 312 to explain why scientific notation is useful in economics. Include the 2004 national debt of $\$ 7,379,100,00,000$ and the U.S. population of $293,700,$ both written in words and in scientific notation, and an explanation of how to find the amount of debt per person with the result written in scientific notation and in standard notation.

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Problem 47

ACT/SAT Which expression is equal to $\frac{\left(2 x^{2}\right)^{3}}{12 x^{4}} ?$
$$
\begin{array}{ll}{\mathbf{A} \frac{x}{2}} & {\mathbf{C} \frac{1}{2 x^{2}}} \\ {\mathbf{B} \frac{2 x}{3}} & {\mathbf{D} \frac{2 x^{2}}{3}}\end{array}
$$

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Problem 48

REVIEW. Four students worked the same math problem. Each student's work is shown below.
$$
\begin{array}{ll}{\frac{\text { Student } \mathrm{F}}{x^{2} x^{-5}=\frac{x^{2}}{x^{5}}}} & {\frac{\text { Student } \mathrm{G}}{x^{2} x^{-5}=\frac{x^{2}}{x^{-5}}}} \\ {=\frac{1}{x^{3}}, x \neq 0} & {=x^{7}, x \neq 0}\end{array}
$$
$$
\begin{array}{l}{\frac{\text { Student } \mathrm{H}}{x^{2} x^{-5}=\frac{x^{2}}{x^{-5}}}} \\ {\quad=x^{-7}, x \neq 0}\end{array}
$$
$$
\begin{array}{l}{\text { Student I }} \\ {\begin{aligned} x^{2} x^{-5} &=\frac{x^{2}}{x^{5}} \\ &=x^{3}, x \neq 0 \end{aligned}}\end{array}
$$
Which is a completely correct solution?
$$
\begin{array}{ll}{\mathbf{F} \text { Student } \mathrm{F}} & {\mathbf{H} \text { Student } \mathrm{H}} \\ {\text { G Student } \mathrm{G}} & {\text { J Student } \mathrm{J}}\end{array}
$$

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Problem 49

Solve each inequality algebraically.
$$
x^{2}-8 x+12<0
$$

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Problem 50

Solve each inequality algebraically.
$$
x^{2}+2 x-86 \geq-23
$$

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Problem 51

Solve each inequality algebraically.
$$
15 x^{2}+4 x+12 \leq 0
$$

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Problem 52

Graph each function.
$$
y=-2(x-2)^{2}+3
$$

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Problem 53

Graph each function.
$$
y=\frac{1}{3}(x+5)^{2}-1
$$

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Problem 54

Graph each function.
$$
y=\frac{1}{2} x^{2}+x+\frac{3}{2}
$$

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Problem 55

Evaluate each determinant.
$$
\left|\begin{array}{rr}{3} & {0} \\ {2} & {-2}\end{array}\right|
$$

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Problem 56

Evaluate each determinant.
$$
\left|\begin{array}{rrr}{1} & {0} & {-3} \\ {2} & {-1} & {4} \\ {-3} & {0} & {2}\end{array}\right|
$$

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Problem 57

Solve each system of equations.
$$
\begin{array}{l}{x+y=5} \\ {x+y+z=4} \\ {2 x-y+2 z=-1}\end{array}
$$

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Problem 58

Solve each system of equations.
$$
\begin{array}{l}{a+b+c=6} \\ {2 a-b+3 c=16} \\ {a+3 b-2 c=-6}\end{array}
$$

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Problem 59

Identify each function as $S$ for step, C for constant, A for absolute value, or $P$ for piecewise.
Graph cannot copy

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Problem 60

Identify each function as $S$ for step, C for constant, A for absolute value, or $P$ for piecewise.
Graph cannot copy

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Problem 61

Identify each function as $S$ for step, C for constant, A for absolute value, or $P$ for piecewise.
Graph cannot copy

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Problem 62

TRANSPORTATION. For Exercises $62-64,$ refer to the graph at the right.
Make a scatter plot of the data, where the horizontal axis is the number of years since $1975 .$

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Problem 63

TRANSPORTATION. For Exercises $62-64,$ refer to the graph at the right.
Write a prediction equation.

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Problem 64

TRANSPORTATION. For Exercises $62-64,$ refer to the graph at the right.
Predict the median age of vehicles on the road in 2015 .

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Problem 65

Solve each equation.
$$
2 x+11=25
$$

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Problem 66

Solve each equation.
$$
-12-5 x=3
$$

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Problem 67

PREREQUISITE SKILL. Use the Distributive Property to find each product.
$$
2(x+y)
$$

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Problem 68

PREREQUISITE SKILL. Use the Distributive Property to find each product.
$$
3(x-z)
$$

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Problem 69

PREREQUISITE SKILL. Use the Distributive Property to find each product.
$$
4(x+2)
$$

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Problem 70

PREREQUISITE SKILL. Use the Distributive Property to find each product.
$$
-2(3 x-5)
$$

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Problem 71

PREREQUISITE SKILL. Use the Distributive Property to find each product.
$$
-5(x-2 y)
$$

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Problem 72

PREREQUISITE SKILL. Use the Distributive Property to find each product.
$$
-3(-y+5)
$$

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