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Algebra 2

Prentice Hall, Basia Kennedy, Dan Ramirez

Chapter 6

Polynomials And Polynomial Functions

Educators


Problem 1

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
7 x+3 x+5
$$

Aditya S.
Numerade Educator

Problem 2

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
5-3 x
$$

Amrita B.
Numerade Educator

Problem 3

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
2 m^{2}-3+7 m
$$

Aditya S.
Numerade Educator

Problem 4

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
-x^{3}+x^{4}+x
$$

Amrita B.
Numerade Educator

Problem 5

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
-4 p+3 p+2 p^{2}
$$

Aditya S.
Numerade Educator

Problem 6

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
5 a^{2}+3 a^{3}+1
$$

Amrita B.
Numerade Educator

Problem 7

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
-x^{5}
$$

Aditya S.
Numerade Educator

Problem 8

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
3+12 x^{4}
$$

Amrita B.
Numerade Educator

Problem 9

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
6 x^{3}-x^{3}
$$

Aditya S.
Numerade Educator

Problem 10

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
7 x^{3}-10 x^{3}+x^{3}
$$

Amrita B.
Numerade Educator

Problem 11

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
4 x+5 x^{2}+8
$$

Aditya S.
Numerade Educator

Problem 12

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
x^{2}-x^{4}+2 x^{2}
$$

Amrita B.
Numerade Educator

Problem 13

Find a cubic model for each set of values.
$$
(-2,-7),(-1,0),(0,1),(1,2),(2,9)
$$

Aditya S.
Numerade Educator

Problem 14

Find a cubic model for each set of values.
$$
(0,-12),(1,10),(2,4),(3,42)
$$

Amrita B.
Numerade Educator

Problem 15

Find a cubic model for each set of values.
$$
(-1,2.5),(0,1),(1,1.5),(2,13)
$$

Karuna R.
Numerade Educator

Problem 16

Find a cubic model for each set of values.
$$
(-3,91),(-2,84),(-1,93),(0,100)
$$

Amrita B.
Numerade Educator

Problem 17

Vital Statistics The data at the right indicate that the life expectancy for residents of the United States has been increasing. Recall that in Chapter 3 you found linear models for this data set.
a. Find quadratic models for the data set.
b. Find cubic models for the data set.
c. Graph each model. Compare the quadratic and cubic models to determine whether one is a better fit.

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Problem 18

Find a cubic model for each function. Then use your model to estimate the value of $y$ when $x=17$ .
$$
(-1,-3),(0,0),(1,-1),(2,0)
$$

Amrita B.
Numerade Educator

Problem 19

Find a cubic model for each function. Then use your model to estimate the value of $y$ when $x=17$ .
$$
(10,0),(11,121),(12,288),(13,507)
$$

Karuna R.
Numerade Educator

Problem 20

Find a cubic model for each function. Then use your model to estimate the value of $y$ when $x=17$ .
$$
(10,500),(14,588),(16,512),(20,0)
$$

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Problem 21

Find a cubic model for each function. Then use your model to estimate the value of $y$ when $x=17$ .
$$
(1,91),(10,95),(20,260),(30,365)
$$

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Problem 22

Find a cubic model for each function. Then use your model to estimate the value of $y$ when $x=17$ .
(TABLE NOT COPY)

Amrita B.
Numerade Educator

Problem 23

Find a cubic model for each function. Then use your model to estimate the value of $y$ when $x=17$ .
(TABLE NOT COPY)

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Problem 24

Open-Ended Write a third-degree polynomial function. Make a table of values and a graph. Find the $x$ - and $y$ -intercepts.

Amrita B.
Numerade Educator

Problem 25

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
8 x-4 x+x^{3}
$$

Aditya S.
Numerade Educator

Problem 26

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
a^{2}+a^{3}-4 a^{4}
$$

Amrita B.
Numerade Educator

Problem 27

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
7
$$

Aditya S.
Numerade Educator

Problem 28

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
2 x(3 x)
$$

Amrita B.
Numerade Educator

Problem 29

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
x^{3}(2+x)
$$

Aditya S.
Numerade Educator

Problem 30

Write each polynomial in standard form. Then classify it by degree and by number of terms.
$$
\frac{3 x^{5}+4 x}{6}
$$

Amrita B.
Numerade Educator

Problem 31

Packaging Design The diagram at the right shows a cologne bottle that consists of a cylindrical base and a hemispherical top.
a. Write an expression for the cylinder's volume.
b. Write an expression for the volume of the hemispherical top.
c. Write a polynomial to represent the total volume.

Karuna R.
Numerade Educator

Problem 32

Writing Explain why cubic functions are useful for interpolating between known data points. Why are they often not reliable for extrapolating data?

Amrita B.
Numerade Educator

Problem 33

Simplify. Classify each result by number of terms.
$$
\left(2 c^{2}+9\right)-\left(3 c^{2}-7\right)
$$

Aditya S.
Numerade Educator

Problem 34

Simplify. Classify each result by number of terms.
$$
\left(-8 d^{3}-7\right)+\left(-d^{3}-6\right)
$$

Amrita B.
Numerade Educator

Problem 35

Simplify. Classify each result by number of terms.
$$
\left(7 x^{2}+8 x-5\right)+\left(9 x^{2}-9 x\right)
$$

Aditya S.
Numerade Educator

Problem 36

Simplify. Classify each result by number of terms.
$$
\left(5 x^{3}-6 x+8\right)-\left(3 x^{3}-9\right)
$$

Amrita B.
Numerade Educator

Problem 37

Simplify. Classify each result by number of terms.
$$
(3 a-2 b)+(6 b-2 a)
$$

Aditya S.
Numerade Educator

Problem 38

Simplify. Classify each result by number of terms.
$$
(4 x-5 y)-(4 x+7 y)
$$

Amrita B.
Numerade Educator

Problem 39

Simplify. Classify each result by number of terms.
$$
\left(3 x^{2}-6 y-1\right)+\left(5 x^{2}+1\right)
$$

Aditya S.
Numerade Educator

Problem 40

Simplify. Classify each result by number of terms.
$$
\left(-a^{2}-3\right)-\left(3 a-a^{2}-5\right)
$$

Amrita B.
Numerade Educator

Problem 41

Simplify. Classify each result by number of terms.
$$
\left(7 x^{3}+9 x^{2}-8 x+11\right)-\left(5 x^{3}-13 x-16\right)
$$

Aditya S.
Numerade Educator

Problem 42

Simplify. Classify each result by number of terms.
$$
\left(-12 x^{3}+5 x-23\right)-\left(4 x^{4}+31-9 x^{3}\right)
$$

Amrita B.
Numerade Educator

Problem 43

Simplify. Classify each result by number of terms.
$$
\left(30 x^{3}-49 x^{2}+7 x\right)+\left(50 x^{3}-75-60 x^{2}\right)
$$

Aditya S.
Numerade Educator

Problem 44

Simplify. Classify each result by number of terms.
$$
\left(-3 x^{3}+7 x^{2}-8\right)-\left(-5 x^{3}+9 x^{2}-8 x+19\right)
$$

Amrita B.
Numerade Educator

Problem 45

Simplify. Classify each result by number of terms.
$$
\left(3 a^{2}-a b-7\right)+\left(5 a^{2}+a b+8\right)-\left(-2 a^{2}+3 a b-9\right)
$$

Aditya S.
Numerade Educator

Problem 46

Find each product. Classify the result by number of terms.
$$
x(2 x)(4 x+1)
$$

Amrita B.
Numerade Educator

Problem 47

Find each product. Classify the result by number of terms.
$$
5 x^{2}(6 x-2)
$$

Aditya S.
Numerade Educator

Problem 48

Find each product. Classify the result by number of terms.
$$
(2 a-5)\left(a^{2}-1\right)
$$

Amrita B.
Numerade Educator

Problem 49

Find each product. Classify the result by number of terms.
$$
b(b-3)^{2}
$$

Aditya S.
Numerade Educator

Problem 50

Find each product. Classify the result by number of terms.
$$
(x-2)^{3}
$$

Amrita B.
Numerade Educator

Problem 51

Find each product. Classify the result by number of terms.
$$
\left(x^{2}+1\right)^{2}
$$

Aditya S.
Numerade Educator

Problem 52

Find each product. Classify the result by number of terms.
$$
(2 x+5)^{3}+1
$$

Amrita B.
Numerade Educator

Problem 53

Find each product. Classify the result by number of terms.
$$
(a-b)^{2}(a+b)
$$

Aditya S.
Numerade Educator

Problem 54

Find each product. Classify the result by number of terms.
$$
(a-1)^{4}
$$

Amrita B.
Numerade Educator

Problem 55

Find each product. Classify the result by number of terms.
$$
(s+3)(4 s-1)(3 s+7)
$$

Aditya S.
Numerade Educator

Problem 56

Find each product. Classify the result by number of terms.
$$
(x+1)(x-1)(x+2)
$$

Amrita B.
Numerade Educator

Problem 57

Find each product. Classify the result by number of terms.
$$
(2 c-3)(2 c+4)(2 c-1)
$$

Aditya S.
Numerade Educator

Problem 58

Find each product. Classify the result by number of terms.
$$
(s+t)(s-t)(s+t)(s-t)
$$

Amrita B.
Numerade Educator

Problem 59

The table shows U.S. energy production for a number of years.
a. Find a linear model, a cubic model, and a quartic model for the data set. Let 0 represent $1960 .$
b. Graph each model. Compare the three models to determine which fits best.
c. Use your answer to part (b) to estimate U.S. energy production in 1997 .

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Problem 60

Geometry Use the formula $V=\frac{\pi h}{3}\left(r^{2}+r s+s^{2}\right)$ to find the volume of the truncated cone. Express your answer in scientific notation with the appropriate number of significant digits.

Amrita B.
Numerade Educator

Problem 61

Critical Thinking Recall that each family of functions has a simplest function called the parent function.
a. Compare the graphs of $y=x^{3}$ and $y=x^{3}+4$ Describe how the graph of $y=x^{3}+4$ relates to the graph of $y=x^{3}$ .
b. Compare the graphs of $y=x^{3}$ and $y=4 x^{3} .$ Describe how the graph of $y=4 x^{3}$ relates to the graph of $y=x^{3}$ .
c. Identify the parent function among the functions in parts (a) and (b).

Karuna R.
Numerade Educator

Problem 62

Which expression is a cubic polynomial?
$$\begin{array}{llll}{\text { A. } x^{3}} & {\text { B. } 3 x+3} & {\text { C. } 2 x^{2}+3 x-1} & {\text { D. } 3 x}\end{array}$$

Amrita B.
Numerade Educator

Problem 63

Which expression is a binomial?
$$\begin{array}{llll}{\text { F. } 2 x} & {\text { G. } \frac{x}{2}} & {\text { H. } 3 x^{2}+2 x+4} & {\text { J. } x-9}\end{array}$$

Aditya S.
Numerade Educator

Problem 64

What is the degree of the polynomial $5 x+4 x^{2}+3 x^{3}-5 x ?$
$$\begin{array}{llll}{\text { A. } 1} & {\text { B. } 2} & {\text { C. } 3} & {\text { D. } 4}\end{array}$$

Amrita B.
Numerade Educator

Problem 65

Which expression is equivalent to $2 x^{4}-3 x+6 ?$
F. $\left(x^{4}-2 x^{2}+3\right)-\left(x^{4}-x^{2}-9\right)$
H. $\left(3 x^{4}-x+3\right)+\left(3-2 x-x^{4}\right)$
G. $2 x^{4}-3(x+6)$
J. $x\left(2 x^{3}-3 x\right)+6$

Karuna R.
Numerade Educator

Problem 66

Simplify $\left(9 x^{3}-4 x+2\right)-\left(x^{3}+3 x^{2}+1\right)$

Amrita B.
Numerade Educator

Problem 67

Simplify $x^{2}\left(3 x^{2}-2 x\right)-3 x^{4} .$ Then name the polynomial by degree and the number of terms.

Aditya S.
Numerade Educator

Problem 68

Why is finding the degree of a polynomial simplified when the polynomial is written in standard form?

Amrita B.
Numerade Educator

Problem 69

Use the discriminant to find the number of real solutions.
$$
3 x^{2}+x-6=0
$$

Aditya S.
Numerade Educator

Problem 70

Use the discriminant to find the number of real solutions.
$$
5 x^{2}-9=0
$$

Amrita B.
Numerade Educator

Problem 71

Use the discriminant to find the number of real solutions.
$$
-x^{2}+2 x-8=0
$$

Aditya S.
Numerade Educator

Problem 72

Graph $f(x)=3 x^{2}-1 .$ Translate the graph right five units and down two units. What is the vertex of the new graph?

Amrita B.
Numerade Educator

Problem 73

Each matrix represents the vertices of a polygon. Translate each figure 3 units left and 2 units down. Express your answer as a matrix.
$$
\left[\begin{array}{rrrr}{4} & {0} & {4} & {8} \\ {-6} & {-1} & {2} & {-1}\end{array}\right]
$$

Aditya S.
Numerade Educator

Problem 74

Each matrix represents the vertices of a polygon. Translate each figure 3 units left and 2 units down. Express your answer as a matrix.
$$
\left[\begin{array}{rrr}{5} & {0} & {-3} \\ {7} & {0} & {2}\end{array}\right]
$$

Amrita B.
Numerade Educator

Problem 75

Each matrix represents the vertices of a polygon. Translate each figure 3 units left and 2 units down. Express your answer as a matrix.
$$
\left[\begin{array}{rrrr}{1} & {2} & {1} & {2} \\ {-1} & {-1} & {-2} & {-2}\end{array}\right]
$$

Aditya S.
Numerade Educator