# Algebra 2

## Educators

Problem 1

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

Problem 2

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

Amrita B.

Problem 3

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

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.

Problem 5

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

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.

Problem 7

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

Problem 8

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

Amrita B.

Problem 9

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

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.

Problem 11

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

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.

Problem 13

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

Problem 14

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

Amrita B.

Problem 15

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

Karuna R.

Problem 16

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

Amrita B.

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.

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.

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.

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.

Problem 25

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

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.

Problem 27

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

Problem 28

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

Amrita B.

Problem 29

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

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.

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.

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.

Problem 33

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

Problem 34

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

Amrita B.

Problem 35

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

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.

Problem 37

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

Problem 38

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

Amrita B.

Problem 39

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

Problem 40

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

Amrita B.

Problem 41

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

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.

Problem 43

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

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.

Problem 45

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

Problem 46

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

Amrita B.

Problem 47

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

Problem 48

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

Amrita B.

Problem 49

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

Problem 50

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

Amrita B.

Problem 51

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

Problem 52

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

Amrita B.

Problem 53

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

Problem 54

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

Amrita B.

Problem 55

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

Problem 56

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

Amrita B.

Problem 57

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

Problem 58

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

Amrita B.

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.

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.

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.

Problem 63

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

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.

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.

Problem 66

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

Amrita B.

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.

Problem 68

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

Amrita B.

Problem 69

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

Problem 70

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

Amrita B.

Problem 71

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

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.

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.
$$\left[\begin{array}{rrrr}{4} & {0} & {4} & {8} \\ {-6} & {-1} & {2} & {-1}\end{array}\right]$$

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.
$$\left[\begin{array}{rrrr}{1} & {2} & {1} & {2} \\ {-1} & {-1} & {-2} & {-2}\end{array}\right]$$