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Section P.3 Polynomials

Before Starting This Section, Review

1 Associative, commutative, and distributive properties (Section p.1 , page 12)
2 Properties of opposites (Section P.1 , page 13)
3 Definition of subtraction (Section P.1 , page 13)
4 Rules for exponents (Section P.2 , page 27)

Objectives

1 Learn polynomial vocabulary.
2 Add and subtract polynomials.
3 Multiply polynomials.
4 Use special-product formulas.

Galileo and Free-Falling Objects

At one time, it was widely believed that heavy objects should fall faster than lighter objects, or, more exactly, that the speed of falling objects ought to be proportional to their weights. So if one object is four times as heavy as another, the heavier object should fall four times as fast as the lighter one. Legend has it that Galileo Galilei (1564–1642) disproved this theory by dropping two balls of equal size, one made of lead and the other of balsa wood, from the top of Italy’s Leaning Tower of Pisa. Both balls are said to have reached the ground at the same instant after they were released simultaneously from the top of the tower.

In fact, such experiments “work” properly only in a vacuum, where objects of different mass do fall at the same rate. In the absence of a vacuum, air resistance may greatly affect the rate at which different objects fall. Whether Galileo’s experiment was actually performed is unknown, but astronaut David R. Scott successfully performed a version of Galileo’s experiment (using a feather and a hammer) on the surface of the moon on August 2, 1971. It turns out that on Earth (ignoring air resistance), regardless of the weight of the object we hold at rest and then drop, the expression 16t2 $16 t^{2}$ gives the distance in feet the object falls in t seconds. If we throw the object down with an initial velocity of $v_{0}$ feet per second, the distance the object travels in t seconds is given by the expression

$16 t^{2} + v_{0} t .$

In Example 1, we evaluate such expressions.

Polynomial Vocabulary

1 Learn polynomial vocabulary.

The height (in feet) of a golf ball above a driving range t seconds after being driven from the tee (at 70 feet per second) is given by the polynomial $70 t - t^{2} .$ After two seconds $(t = 2),$ the ball is $70 (2) - {(2)}^{2} = 136$ feet above the ground. Polynomials such as $70 t - t^{2}$ appear frequently in applications.

We begin by reviewing the basic vocabulary of polynomials. A monomial is the simplest polynomial; it contains one term. In the variable x it has the form $a x^{k},$ where a is a constant and k is either a positive integer or zero. The constant a is called the coefficient of the monomial. For $a \neq 0,$ the integer k is called the degree of the monomial. If $a = 0,$ the monomial is 0 and has no degree. Any variable may be used in place of x to form a monomial in that variable.

The following are examples of monomials.

$2 x^{5}$	The coefficient is 2, and the degree is 5.
$- 3 x^{2}$	The coefficient is $- 3,$ and the degree is 2.
$- 7$	$- 7 = - 7 (1) = - 7 x^{0} .$ The coefficient is $- 7,$ and the degree is 0.
8`x`	The coefficient is 8, and the degree is $1 (x = x^{1}) .$
$x^{12}$	The coefficient is $1 (x^{12} = 1 \cdot x^{12}),$ and the degree is 12.
$- x^{4}$	The coefficient is $- 1 (- x^{4} = (- 1) \cdot x^{4}),$ and the degree is 4.

The expression $5 x^{- 3}$ is not a monomial because the exponent on x is negative.

Two monomials in the same variable with the same degree can be combined into one monomial using the distributive property. For example, $- 3 x^{5}$ and $9 x^{5}$ can be added: $- 3 x^{5} + 9 x^{5} = (- 3 + 9) x^{5} = 6 x^{5} .$ Or $- 3 x^{5}$ and $9 x^{5}$ can be subtracted: $- 3 x^{5} - 9 x^{5} = (- 3 - 9) x^{5} = - 12 x^{5} .$ Two monomials in the same variable with the same degree are called like terms.

The symbols $a_{0}, a_{1}, a_{2}, \dots a_{n}$ in the general notation for a polynomial

$a_{n} x^{n} + a_{n - 1} x^{n - 1} +^{} \dots + a_{2} x^{2} + a_{1} x + a_{0}$

are just constants; the numbers to the lower right of a are called subscripts. The notation $a_{2}$ is read “a sub 2”, $a_{1}$ is read “a sub 1”, and $a_{0}$ is read “a sub 0”. This type of notation is used when a large or indefinite number of constants are required. The terms of the polynomial $5 x^{2} + 3 x + 1$ are $5 x^{2},$ 3x, and 1; the coefficients are 5, 3, and 1; and its degree is 2.

Note that the terms in the general form for a polynomial are connected by the addition symbol $+ .$ How do we handle $5 x^{2} - 3 x + 1 ?$ Because subtraction is defined in terms of addition, we rewrite

$5 x^{2} - 3 x + 1 = 5 x^{2} + (- 3) x + 1$

and see that the terms of $5 x^{2} - 3 x + 1$ are $5 x^{2},$ $- 3 x,$ and 1 and that the coefficients are $5, - 3,$ and 1. (The degree is 2.) Once we recognize this fact, we no longer rewrite the polynomial with the $+$ sign separating terms; we just remember that the “sign” is part of both the term and the coefficient. Polynomials and their properties will be discussed in more detail in Chapter 3.

Polynomials can be classified according to the number of terms they have. Polynomials with one term are called monomials, polynomials with two unlike terms are called bino-mials, and polynomials with three unlike terms are called trinomials. Polynomials with more than three unlike terms do not have special names.

By agreement, the only polynomial that has no degree is the zero polynomial, which results when all of the coefficients are 0. It is easiest to find the degree of a polynomial when it is written in descending order—that is, when the exponents decrease from left to right. In this case, the degree is the exponent of the leftmost term. A polynomial written in descending order is said to be in standard form.

Example 1 Examining Free-Falling Objects

In our introductory discussion about Galileo and free-falling objects, we mentioned two well-known polynomials.

$16 t^{2}$ gives the distance in feet a free-falling object falls in t seconds.
With $v_{0} = 10,$ the expression $16 t^{2} + 10 t$ gives the distance an object falls in t seconds when it is thrown down with an initial velocity of 10 feet per second. Use these polynomials to
1. Find how far a wallet dropped from the 86th-floor observatory of the Empire State Building will fall in 5 seconds.
2. Find how far a quarter thrown down with an initial velocity of 10 feet per second from a hot air balloon will travel after 5 seconds.

Solution

The value of $16 t^{2}$ for $t = 5$ is $16 {(5)}^{2} = 400 .$ The wallet falls 400 feet.
The value of $16 t^{2} + 10 t$ for $t = 5$ is $16 {(5)}^{2} + 10 (5) = 450 .$ The quarter travels 450 feet on its downward path after five seconds.

Practice Problem 1

If $16 t^{2} + 15 t$ gives the distance an object falls in t seconds when it is thrown down at an initial velocity of 15 feet per second, find how far a quarter thrown down with an initial velocity of 15 feet per second from a hot air balloon has traveled after 7 seconds.

Adding and Subtracting Polynomials

2 Add and subtract polynomials.

Like monomials, polynomials are added and subtracted by combining like terms. By convention, we write polynomials in standard form.

Example 2 Adding Polynomials

Find the sum of the polynomials

$- 4 x^{3} + 5 x^{2} + 7 x - 2 and 6 x^{3} - 2 x^{2} - 8 x - 5 .$

Horizontal Method: Group like terms and then combine them.

$\begin{array}{l} (- 4 x^{3} + 5 x^{2} + 7 x - 2) + (6 x^{3} - 2 x^{2} - 8 x - 5) \\ = (- 4 x^{3} + 6 x^{3}) + (5 x^{2} - 2 x^{2}) + (7 x - 8 x) + (- 2 - 5) & Group like terms . \\ = 2 x^{3} + 3 x^{2} - x - 7 & Combine like terms . \end{array}$

A second method for adding polynomials is to arrange them in columns so that like terms appear in the same column.

Column Method:

$\begin{aligned} - 4 x^{3} + 5 x^{2} + 7 x - 2 \\ \underline{(+) 6 x^{3} - 2 x^{2} - 8 x - 5} \\ 2 x^{3} + 3 x^{2} - x - 7 & (answer) \end{aligned}$

Practice Problem 2

Find the sum of

$7 x^{3} + 2 x^{2} - 5 and - 2 x^{3} + 3 x^{2} + 2 x + 1 .$

Example 3 Subtracting Polynomials

Find the difference of the polynomials

$(9 x^{4} - 6 x^{3} + 4 x^{2} - 1) - (2 x^{4} - 11 x^{3} + 5 x + 3) .$

Horizontal Method: First, using the definition of subtraction, change the sign of each term in the second polynomial. Then add the resulting polynomials by grouping like terms and combining them.

$\begin{array}{l} (9 x^{4} - 6 x^{3} + 4 x^{2} - 1) - (2 x^{4} - 11 x^{3} + 5 x + 3) \\ = (9 x^{4} - 6 x^{3} + 4 x^{2} - 1) + (- 2 x^{4} + 11 x^{3} - 5 x - 3) \\ = (9 x^{4} - 2 x^{4}) + (- 6 x^{3} + 11 x^{3}) + 4 x^{2} - 5 x + (- 1 - 3) & Group like terms . \\ = 7 x^{4} + 5 x^{3} + 4 x^{2} - 5 x - 4 & Combine like terms . \end{array}$

The second method for finding the difference of polynomials requires arranging them in columns so that like terms appear in the same column.

Column Method: To find the difference, we again change the sign of each term in the second polynomial and then add.

$\begin{aligned} 9 x^{4} - 6 x^{3} + 4 x^{2} - 1 \\ \underline{(+) - 2 x^{4} + 11 x^{3} - 5 x - 3} & Change signs and then add . \\ 7 x^{4} + 5 x^{3} + 4 x^{2} - 5 x - 4 & (answer) \end{aligned}$

Practice Problem 3

Find the difference of

$3 x^{4} - 5 x^{3} + 2 x^{2} + 7 and - 2 x^{4} + 3 x^{2} + x - 5 .$

Multiplying Polynomials

3 Multiply polynomials.

Previously, we multiplied two monomials, such as $- 3 x^{2}$ and $5 x^{3},$ by using the product rule for exponents: $a^{m} \cdot a^{n} = a^{m + n} .$ We multiply a monomial and a polynomial by combining the product rule and the distributive property.

Example 4 Multiplying a Monomial and a Trinomial

Multiply $4 x^{3}$ and $3 x^{2} - 5 x + 7 .$

$\begin{array}{l} 4 x^{3} (3 x^{2} - 5 x + 7) \\ = (4 x^{3}) (3 x^{2}) + (4 x^{3}) (- 5 x) + (4 x^{3}) (7) & Distributive property \\ = (4 \cdot 3) x^{3 + 2} + 4 (- 5) x^{3 + 1} + (4 \cdot 7) x^{3} & Multiply monomials . \\ = 12 x^{5} - 20 x^{4} + 28 x^{3} & Simplify . \end{array}$

Practice Problem 4

Multiply $- 2 x^{3}$ and $4 x^{2} + 2 x - 5 .$

We can use the distributive property repeatedly to multiply any two polynomials. As with addition, there are two methods.

Example 5 Multiplying a Binomial and a Trinomial

Multiply $4 x^{2} + 3 x$ and $x^{2} + 2 x - 3 .$

Horizontal Method: We treat the second polynomial as a single term and use the distributive property, $(a + b) c = a c + b c,$ to multiply it by each of the two terms in the first polynomial.

$\begin{array}{l} (4 x^{2} + 3 x) (x^{2} + 2 x - 3) \\ = 4 x^{2} (x^{2} + 2 x - 3) + 3 x (x^{2} + 2 x - 3) & Distributive property \\ = 4 x^{2} (x^{2}) + 4 x^{2} (2 x) + 4 x^{2} (- 3) + 3 x (x^{2}) + 3 x (2 x) + 3 x (- 3) & Distributive property \\ = 4 x^{4} + 8 x^{3} - 12 x^{2} + 3 x^{3} + 6 x^{2} - 9 x & Product rule \\ = 4 x^{4} + (8 x^{3} + 3 x^{3}) + (- 12 x^{2} + 6 x^{2}) - 9 x & Group like terms . \\ = 4 x^{4} + 11 x^{3} - 6 x^{2} - 9 x & Combine like terms . \end{array}$

The column method resembles the multiplication of two positive integers written in column form.

Column Method:

Practice Problem 5

Multiply $5 x^{2} + 2 x$ and $- 2 x^{2} + x - 7 .$

The basic multiplication rule for polynomials can be stated as follows.

Special Products

4 Use special-product formulas.

Particular polynomial products called special products occur frequently enough to deserve special attention. We introduce a very useful method, called FOIL, for multiplying two binomials. Notice that

$\begin{array}{rcll} (x + 2) (x - 7) & = & x (x - 7) + 2 (x - 7) & Distributive property \\ = & x^{2} - 7 x + 2 x - 14 & Distributive property \\ = & x^{2} - 5 x - 14 & Combine like terms . \end{array}$

Now consider the relationship between the terms of the factors in $(x + 2) (x - 7)$ and the second line, $x^{2} - 7 x + 2 x - 14,$ in the computation of the product.

Example 6 Using the FOIL Method

Use the FOIL method to find the following products.

$\begin{array}{rcll} (2 x + 3) (x - 1) & = & \overset{F}{(2 x) (x)} + \overset{O}{(2 x) (- 1)} + \overset{I}{(3) (x)} + \overset{L}{(3) (- 1)} \\ = & 2 x^{2} - 2 x + 3 x - 3 \\ = & 2 x^{2} + x - 3 & Combine like terms . \end{array}$
$\begin{array}{rcll} (3 x - 5) (4 x - 6) & = & \overset{F}{(3 x) (4 x)} + \overset{O}{(3 x) (- 6)} + \overset{I}{(- 5) (4 x)} + \overset{L}{(- 5) (- 6)} \\ = & 12 x^{2} - 18 x - 20 x + 30 \\ = & 12 x^{2} - 38 x + 30 & Combine like terms . \end{array}$
$\begin{array}{rcll} (x + a) (x + b) & = & \overset{F}{x \cdot x} + \overset{O}{x \cdot b} + \overset{I}{a \cdot x} + \overset{L}{a \cdot b} \\ = & x^{2} + b x + a x + a \cdot b & x b = b x (commutative property) \\ = & x^{2} + (b + a) x + a b & Combine like terms . \end{array}$

Practice Problem 6

Use FOIL to find each product.
1. $(4 x - 1) (x + 7)$
2. $(3 x - 2) (2 x - 5)$

Squaring a Binomial Sum or Difference

We can find a general formula for the square of any binomial sum ${(A + B)}^{2}$ by using FOIL.

$\begin{array}{rcll} {(A + B)}^{2} = (A + B) (A + B) & = & \overset{F}{A \cdot A} + \overset{O}{A \cdot B} + \overset{I}{B \cdot A} + \overset{L}{B \cdot B} \\ = & A^{2} + A B + A B + B^{2} & A B = B A \\ = & A^{2} + 2 A B + B^{2} & A B + A B = 2 A B \end{array}$

A formula such as ${(A + B)}^{2} = A^{2} + 2 A B + B^{2}$ can be used two ways.

Example 7 Finding the Square of a Binomial Sum

Find ${(2 x + 3)}^{2} .$

Solution

Pattern Method: In ${(2 x + 3)}^{2},$ the first term is 2x and the second term is 3. Rewrite the formula as

$\begin{array}{l} \begin{array}{c} \begin{array}{l} Square of \\ sum \end{array} & = & \begin{array}{l} Square of \\ first term \end{array} & + & 2 \begin{array}{l} Product of first \\ and second terms \end{array} & + & \begin{array}{l} Square of \\ second term \end{array} \\ {(2 x + 3)}^{2} & = & {(2 x)}^{2} & + & 2 \cdot (2 x) \cdot 3 & + & 3^{2} \end{array} \\ \begin{array}{l} = 2^{2} x^{2} + 2 \cdot 2 \cdot 3 \cdot x + 3^{2} & Group numerical factors . \\ = 4 x^{2} + 12 x + 9 & Simplify . \end{array} \end{array}$

Substitution Method: In the formula ${(A + B)}^{2} = A^{2} + 2 A B + B^{2},$ substitute $2 x = A$ and $3 = B .$ Then

$\begin{array}{rcll} {(2 x + 3)}^{2} & = & {(2 x)}^{2} + 2 \cdot (2 x) \cdot 3 + 3^{2} \\ = & 2^{2} x^{2} + 2 \cdot 2 \cdot 3 \cdot x + 3^{2} & Group numerical factors . \\ = & 4 x^{2} + 12 x + 9 & Simplify . \end{array}$

Practice Problem 7

Find ${(3 x + 2)}^{2} .$

The definition of subtraction allows us to write any difference $A - B$ as the sum $A + (- B) .$ Because $A - B = A + (- B),$ we can use the formula for the square of a binomial sum to find a formula for the square of a binomial difference ${(A - B)}^{2} .$

$\begin{array}{rcll} {(A - B)}^{2} & = & {[A + (- B)]}^{2} & First term = A; second term = - B \\ = & A^{2} + 2 A (- B) + {(- B)}^{2} \\ = & A^{2} - 2 A B + B^{2} & 2 A (- B) = - 2 A B; {(- B)}^{2} = B^{2} \end{array}$

Warning

$\begin{array}{c} Incorrect & Correct \\ \begin{array}{rcl} {(x + y)}^{2} & = & x + y^{2} \\ {(x + y)}^{2} & = & x^{2} + 2 x y + y^{2} \\ {(x - y)}^{2} & = & x^{2} + y^{2} \end{array} & \begin{array}{rcl} {(x - y)}^{2} & = & x^{2} - 2 x y + y^{2} \\ (x + 4) (x + 5) & = & x + 4 x + 5 \\ (x + 4) (x + 5) & = & x^{2} + 9 x + 20 \end{array} \end{array}$

The Product of the Sum and Difference of Terms

We leave it for you to use FOIL to verify the formula for finding the product of the sum and difference of terms.

Example 8 Finding the Product of the Sum and Difference of Terms

Find the product $(3 x + 4) (3 x - 4) .$

Solution

We use the substitution method; substitute $3 x = A$ and $4 = B$ in the formula $(A + B) (A - B) = A^{2} - B^{2} . Then$

$\begin{array}{rcll} (3 x + 4) (3 x - 4) & = & {(3 x)}^{2} - 4^{2} \\ = & 9 x^{2} - 16 & {(3 x)}^{2} = 3^{2} x^{2} = 9 x^{2} \end{array}$

Practice Problem 8

Find the product $(1 - 2 x) (1 + 2 x) .$

The special-products formulas, such as the formulas for squaring a binomial sum or difference, are used often and should be memorized. However, you should be able to derive them from FOIL, if you forget them. We list several of these formulas next.

Special-Product Formulas

A and B represent any algebraic expressions.

Formula	Example
Product giving a difference of squares $(A + B) (A - B) = A^{2} - B^{2}$	$\begin{array}{rcl} (5 x + 2) (5 x - 2) & = & {(5 x)}^{2} - 2^{2} \\ = & 25 x^{2} - 4 \end{array}$
Squaring a binomial sum or difference $\begin{array}{rcl} {(A + B)}^{2} & = & A^{2} + 2 A B + B^{2} \\ {(A - B)}^{2} & = & A^{2} - 2 A B + B^{2} \end{array}$	$\begin{array}{rcl} {(3 x + 2)}^{2} & = & {(3 x)}^{2} + 2 (3 x) (2) + 2^{2} \\ = & 9 x^{2} + 12 x + 4 \\ {(2 x - 5)}^{2} & = & {(2 x)}^{2} - 2 (2 x) (5) + 5^{2} \\ = & 4 x^{2} - 20 x + 25 \end{array}$
Cubing a binomial sum or difference $\begin{array}{rcl} {(A + B)}^{3} & = & A^{3} + 3 A^{2} B + 3 A B^{2} + B^{3} \\ {(A - B)}^{3} & = & A^{3} - 3 A^{2} B + 3 A B^{2} - B^{3} \end{array}$	$\begin{array}{rcl} {(x + 5)}^{3} & = & x^{3} + 3 x^{2} (5) + 3 x {(5)}^{2} + 5^{3} \\ = & x^{3} + 15 x^{2} + 75 x + 125 \\ {(x - 4)}^{3} & = & x^{3} - 3 x^{2} (4) + 3 x {(4)}^{2} - 4^{3} \\ = & x^{3} - 12 x^{2} + 48 x - 64 \end{array}$
Products giving a sum or difference of cubes $\begin{array}{rcl} (A + B) (A^{2} - A B + B^{2}) & = & A^{3} + B^{3} \\ (A - B) (A^{2} + A B + B^{2}) & = & A^{3} - B^{3} \end{array}$	$\begin{array}{rcl} (x + 2) (x^{2} - 2 x + 4) & = & x^{3} + 2^{3} = x^{3} + 8 \\ (x - 3) (x^{2} + 3 x + 9) & = & x^{3} - 3^{3} = x^{3} - 27 \end{array}$

To this point, we have investigated polynomials only in a single variable. We next extend our previous vocabulary to discuss polynomials in more than one variable. Any product of a constant and two or more variables, each raised to whole-number powers, is a monomial in those variables. The constant is called the coefficient of the monomial, and the degree of the monomial is the sum of all the exponents appearing on its variables. For example, the monomial $5 x^{3} y^{4}$ is of degree $3 + 4 = 7$ with coefficient 5.

Example 9 Multiplying Polynomials in Two Variables

Multiply.

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

Solution

$\begin{array}{rcll} (7 a + 5 b) (7 a - 5 b) & = & {(7 a)}^{2} - {(5 b)}^{2} & (A + B) (A - B) = A^{2} - B^{2} \\ = & 49 a^{2} - 25 b^{2} & Simplify . \end{array}$
$\begin{array}{rcll} {(2 x + 3 y)}^{3} & = & {(2 x)}^{3} + 3 {(2 x)}^{2} (3 y) + 3 (2 x) {(3 y)}^{2} + {(3 y)}^{3} & {(A + B)}^{3} = A^{3} + 3 A^{2} B + 3 A B^{2} + B^{3} \\ = & 2^{3} x^{3} + 3 (2^{2} x^{2}) (3 y) + 3 (2 x) (3^{2} y^{2}) + 3^{3} y^{3} & Power-of-a-product rule \\ = & 8 x^{3} + 36 x^{2} y + 54 x y^{2} + 27 y^{3} & Simplify . \end{array}$

Practice Problem 9

Multiply.
1. $(x + 2 y) (x - 2 y)$
2. ${(2 x - y)}^{3}$

Section P.3 Exercises

Concepts and Vocabulary

The polynomial $- 3 x^{7} + 2 x^{2} - 9 x + 4$ has leading coefficient and degree .
When a polynomial is written so that the exponents in each term decrease from left to right, it is said to be in form.
When a polynomial in x of degree 3 is added to a polynomial in x of degree 4, the resulting polynomial has degree .
When a polynomial in x of degree 3 is multiplied by a polynomial in x of degree 4, the resulting polynomial has degree .
True or False. When FOIL is used, the terms 7 and x are called the inside terms of the product $(4 x + 7) (x - 2) .$
True or False. There are values of A and B for which ${(A + B)}^{2} = A^{2} + B^{2} .$
True or False. The expression $x^{3} - 3 \sqrt{x} + 7$ is a polynomial of degree 3.
True or False. The expression $x^{5} - \sqrt{2} x^{3} + 11$ is a polynomial of degree 5.

Building Skills

In Exercises 9–16, determine whether the given expression is a polynomial. If it is, write it in standard form.

$1 + x^{2} + 2 x$
$x - \frac{1}{x}$
$x^{- 2} + 3 x + 5$
$3 x^{4} + x^{7} + 3 x^{5} - 2 x + 1$
$x^{2} - 2 | x | + 4$
$x^{3} + 5 \sqrt{x} + 1$
$5 x^{3} - | π - 7 | x + 11$
$2 x^{2} - \sqrt{3} x + 4$

In Exercises 17–20, find the degree and list the terms of the polynomial.

$7 x + 3$
$- 3 x^{2} + 7$
$x^{2} - x^{4} + 2 x - 9$
$x + 2 x^{3} + 9 x^{7} - 21$

In Exercises 21–30, perform the indicated operations. Write the resulting polynomial in standard form.

$(x^{3} + 2 x^{2} - 5 x + 3) + (- x^{3} + 2 x - 4)$
$(x^{3} - 3 x + 1) + (x^{3} - x^{2} + x - 3)$
$(2 x^{3} - x^{2} + x - 5) - (x^{3} - 4 x + 3)$
$(- x^{3} + 2 x - 4) - (x^{3} + 3 x^{2} - 7 x + 2)$
$(- 2 x^{4} + 3 x^{2} - 7 x) - (8 x^{4} + 6 x^{3} - 9 x^{2} - 17)$
$3 (x^{2} - 2 x + 2) + 2 (5 x^{2} - x + 4)$
$- 2 (3 x^{2} + x + 1) + 6 (- 3 x^{2} - 2 x - 2)$
$2 (5 x^{2} - x + 3) - 4 (3 x^{2} + 7 x + 1)$
$(3 y^{3} - 4 y + 2) + (2 y + 1) - (y^{3} - y^{2} + 4)$
$(5 y^{2} + 3 y - 1) - (y^{2} - 2 y + 3) + (2 y^{2} + y + 5)$

In Exercises 31–78, perform the indicated operations.

$6 x (2 x + 3)$
$7 x (3 x - 4)$
$(x + 1) (x^{2} + 2 x + 2)$
$(x - 5) (2 x^{2} - 3 x + 1)$
$(3 x - 2) (x^{2} - x + 1)$
$(2 x + 1) (x^{2} - 3 x + 4)$
$(x + 1) (x + 2)$
$(x + 2) (x + 3)$
$(3 x + 2) (3 x + 1)$
$(x + 3) (2 x + 5)$
$(- 4 x + 5) (x + 3)$
$(- 2 x + 1) (x - 5)$
$(3 x - 2) (2 x - 1)$
$(x - 1) (5 x - 3)$
$(2 x - 3 a) (2 x + 5 a)$
$(5 x - 2 a) (x + 5 a)$
${(x + 2)}^{2} - x^{2}$
${(x - 3)}^{2} - x^{2}$
${(4 x + 1)}^{2}$
${(3 x + 2)}^{2}$
${(x + \frac{3}{4})}^{2}$
${(x + \frac{2}{5})}^{2}$
${(3 x - 4 y)}^{2}$
${(2 x + 5 y)}^{2}$
${(x^{2} + 2)}^{2}$
${(3 - x^{2})}^{2}$
${(\frac{x}{2} + \frac{2}{x})}^{2}$
${(\frac{3}{y} - \frac{y}{3})}^{2}$
${(x + 2)}^{3}$
${(2 x - 1)}^{3}$
${(x + 3)}^{3} - x^{3}$
${(x - 2)}^{3} - x^{3}$
${(x + 2 y)}^{3}$
${(2 x + 3 y)}^{3}$
$(5 + 2 x) (5 - 2 x)$
$(3 - 4 x) (3 + 4 x)$
$(2 x + 3 y) (2 x - 3 y)$
$(5 x - 2 y) (5 x + 2 y)$
$(x + \frac{1}{x}) (x - \frac{1}{x})$
$(\frac{y}{2} - \frac{2}{y}) (\frac{y}{2} + \frac{2}{y})$
$(x^{2} + 3) (x^{2} - 3)$
$(x^{3} - 2) (x^{3} + 2)$
$(2 x - 3) (x^{2} - 3 x + 5)$
$(x - 2) (x^{2} - 4 x - 3)$
$(1 + y) (1 - y + y^{2})$
$(y + 4) (y^{2} - 4 y + 16)$
$(x - 6) (x^{2} + 6 x + 36)$
$(x - 1) (x^{2} + x + 1)$

In Exercises 79–88, perform the indicated operations.

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

In Exercises 89–94, use $a^{2} + b^{2} = {(a + b)}^{2} - 2 a b = {(a - b)}^{2} + 2 a b .$

If $x + y = 4$ and $x y = 3,$ find the value of $x^{2} + y^{2} .$
If $x - y = 3$ and $x y = 2,$ find the value of $x^{2} + y^{2} .$
If $x + \frac{1}{x} = 3,$ find the value of
1. $x^{2} + \frac{1}{x^{2}} .$
2. $x^{4} + \frac{1}{x^{4}} .$
If $x - \frac{1}{x} = 2,$ find the value of
1. $x^{2} + \frac{1}{x^{2}} .$
2. $x^{4} + \frac{1}{x^{4}} .$
If $3 x + 2 y = 12$ and $x y = 6,$ find the value of $9 x^{2} + 4 y^{2} .$
If $3 x - 7 y = 10$ and $x y = - 1,$ find the value of $9 x^{2} + 49 y^{2} .$

Applying the Concepts

Ticket prices. Theater ticket prices (in dollars) x years after 2006 are described by the polynomial $- 0.025 x^{2} + 0.44 x + 4.28 .$ Find the value of this polynomial for $x = 6$ and state the result as the ticket price for a specific year.
Box-office grosses. Theater box-office grosses (in billions of dollars) x years after 2008 are described by the polynomial $0.035 x^{2} + 0.15 x + 5.17 .$ Find the value of this polynomial for 2012 and state the result as box-office gross for a specific year.
Paper shredding. A business that shreds paper products finds that it costs $0.1 x^{2} + x + 50$ dollars to serve x customers. What does it cost to serve 40 customers?
Gift baskets. A company will produce x $(for x \geq 10)$ gift baskets of wine, meat, cheese, and crackers at a cost of ${(x - 10)}^{2} + 5 x$ dollars per basket. What is the per basket cost for 15 baskets?

In Exercises 99 and 100, use the fact (from Example 1) that an object thrown down with an initial velocity of $v_{0}$ feet per second will travel $16 t^{2} + v_{0} t$ feet in t seconds.

Free fall. A sandwich is thrown from a helicopter with an initial downward velocity of 20 feet per second. How far has the sandwich fallen after five seconds?
Free fall. A ring is thrown off the Empire State Building with an initial downward velocity of 10 feet per second. How far has the ring fallen after two seconds?
Cruise ship revenue. A cruise ship decides to reduce ticket prices to its theater by x dollars from the current price of $22.50.
1. Write a polynomial that gives the new price after the x-dollar deduction.
2. The revenue is the number of tickets sold times the price of each ticket. If 30 tickets were sold when the price was $22.50 and 10 additional tickets are sold for each dollar the price is reduced (all tickets are sold at the same price), write a polynomial that gives the revenue in terms of x when the original price is reduced by x dollars.
Used-car rentals. A dealer who rents used cars wants to increase the monthly rent from the current $250 in n increases of $10.
1. Write a polynomial that gives the new price after n increases of $10.
2. The revenue is the number of cars rented times the monthly rent for each car. If 50 cars can be rented at $250 per month and 2 fewer cars can be rented for each $10 rent increase, write a polynomial that gives the monthly revenue in terms of n.

Beyond the Basics

Show that ${(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a .$

In Exercises 104–108 use Exercise 103.

Find ${(x^{2} + x + 1)}^{2} .$
Simplify ${(2 x + y - z)}^{2} - {(2 x - y + z)}^{2} .$
If $a + b + c = 8$ and $a b + b c + c a = 12,$ find the value of $a^{2} + b^{2} + c^{2} .$
If $x + y + z = 12$ and $x^{2} + y^{2} + z^{2} = 44,$ find the value of $x y + y z + z x .$
If $x^{2} + y^{2} + z^{2} = 64$ and $x y + y z + z x = 18,$ find the value of $x + y + z .$
Show that $(a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a) = a^{3} + b^{3} + c^{3} - 3 a b c .$
Use Exercise 109 to show that if $a + b + c = 0,$ then $a^{3} + b^{3} + c^{3} = 3 a b c .$

In Exercises 111–114 use Exercises 103 and 110 where applicable.

Show that ${(x - y)}^{3} + {(y - z)}^{3} + {(z - x)}^{3} = 3 (x - y) (y - z) (z - x) .$
Show that ${(2 x - 3 y)}^{3} + {(3 y - 5 z)}^{3} + {(5 z - 2 x)}^{3} = 3 (2 x - 3 y) (3 y - 5 z) (5 z - 2 x) .$
If $a + b + c = 8$ and $a b + b c + c a = 19,$ find the value of $a^{3} + b^{3} + c^{3} - 3 a b c .$
If $a + b + c = 9$ and $a b + b c + c a = 11,$ find the value of $a^{3} + b^{3} + c^{3} - 3 a b c .$

Getting Ready for the Next Section

In Exercises 115–124 find the missing integers in each row. In Exercises 119–124 choose a and b with $| a | < | b | .$

	`a`	`b`	$a + b$	`ab`
115.	3	4
116.	$- 3$	5
117.		2	6	8
118.		$- 5$	$- 2$	$- 15$
119.			7	10
120.			8	15
121.			2	$- 35$
122.			$- 2$	$- 35$
123.			$- 5$	6
124.			5	6

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Table of Contents for Section P.3 Polynomials

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Table of Contents for
Section P.3 Polynomials