Contents

Preface
1Foundations
- Introduction
- 1.1Use the Language of Algebra
- 1.2Integers
- 1.3Fractions
- 1.4Decimals
- 1.5Properties of Real Numbers
- Chapter Review
  Key Terms
  Key Concepts
- Exercises
  Review Exercises
  Practice Test
2Solving Linear Equations
3Graphs and Functions
4Systems of Linear Equations
5Polynomials and Polynomial Functions
6Factoring
7Rational Expressions and Functions
8Roots and Radicals
9Quadratic Equations and Functions
10Exponential and Logarithmic Functions
11Conics
12Sequences, Series and Binomial Theorem
- Introduction
- 12.1Sequences
- 12.2Arithmetic Sequences
- 12.3Geometric Sequences and Series
- 12.4Binomial Theorem
- Chapter Review
  Key Terms
Answer Key
Index

Add and Subtract Polynomials

Learning Objectives

By the end of this section, you will be able to:

Determine the degree of polynomials
Add and subtract polynomials
Evaluate a polynomial function for a given value
Add and subtract polynomial functions

Be Prepared 5.1

Before you get started, take this readiness quiz.

Simplify: $3 x^{2} + 3 x + 1 + 8 x^{2} + 5 x + 5 . 3 x^{2} + 3 x + 1 + 8 x^{2} + 5 x + 5 .$
If you missed this problem, review Example 1.7.
Subtract: $(5 n + 8) - (2 n - 1) . (5 n + 8) - (2 n - 1) .$
If you missed this problem, review Example 1.5.
Evaluate: $4 x y^{2} 4 x y^{2}$ when $x = −2 x = −2$ and $y = 5 . y = 5 .$
If you missed this problem, review Example 1.21.

Determine the Degree of Polynomials

We have learned that a term is a constant or the product of a constant and one or more variables. A monomial is an algebraic expression with one term. When it is of the form $a x^{m}, a x^{m},$ where a is a constant and m is a whole number, it is called a monomial in one variable. Some examples of monomial in one variable are. Monomials can also have more than one variable such as and $−4 a^{2} b^{3} c^{2} . −4 a^{2} b^{3} c^{2} .$

Monomial

A monomial is an algebraic expression with one term.

A monomial in one variable is a term of the form $a x^{m}, a x^{m},$ where a is a constant and m is a whole number.

A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.

Polynomials

polynomial—A monomial, or two or more algebraic terms combined by addition or subtraction is a polynomial.

monomial—A polynomial with exactly one term is called a monomial.

binomial—A polynomial with exactly two terms is called a binomial.

trinomial—A polynomial with exactly three terms is called a trinomial.

Here are some examples of polynomials.

Polynomial	$y + 1 y + 1$	$4 a^{2} - 7 a b + 2 b^{2} 4 a^{2} - 7 a b + 2 b^{2}$	$4 x^{4} + x^{3} + 8 x^{2} - 9 x + 1 4 x^{4} + x^{3} + 8 x^{2} - 9 x + 1$
Monomial	14	$8 y^{2} 8 y^{2}$	$−9 x^{3} y^{5} −9 x^{3} y^{5}$	$−13 a^{3} b^{2} c −13 a^{3} b^{2} c$
Binomial	$a + 7 b a + 7 b$	$4 x^{2} - y^{2} 4 x^{2} - y^{2}$	$y^{2} - 16 y^{2} - 16$	$3 p^{3} q - 9 p^{2} q 3 p^{3} q - 9 p^{2} q$
Trinomial	$x^{2} - 7 x + 12 x^{2} - 7 x + 12$	$9 m^{2} + 2 m n - 8 n^{2} 9 m^{2} + 2 m n - 8 n^{2}$	$6 k^{4} - k^{3} + 8 k 6 k^{4} - k^{3} + 8 k$	$z^{4} + 3 z^{2} - 1 z^{4} + 3 z^{2} - 1$

Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial, binomial, and trinomial when referring to these special polynomials and just call all the rest polynomials.

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

A monomial that has no variable, just a constant, is a special case. The degree of a constant is 0.

Degree of a Polynomial

The degree of a term is the sum of the exponents of its variables.

The degree of a constant is 0.

The degree of a polynomial is the highest degree of all its terms.

Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

Let's start by looking at a monomial. The monomial $8 a b^{2} 8 a b^{2}$ has two variables a and b. To find the degree we need to find the sum of the exponents. The variable a doesn't have an exponent written, but remember that means the exponent is 1. The exponent of b is 2. The sum of the exponents, $1 + 2, 1 + 2,$ is 3 so the degree is 3.

The polynomial is 8 a b squared. The exponents of the variables are 1 and 2 so the degree of the monomial is 1 plus 2 which equals 3.

Here are some additional examples.

Monomial examples: 14 has degree 0, 8 a b squared has degree 3, negative 9 x cubed y to the fifth power has degree 8, negative 13 a has degree 1. Binomial examples: The terms in h plus 7 have degree 1 and 0 so the degree of the whole polynomial is 1. The terms in 7 b squared minus 3 b have degree 2 and 1 so the degree of the whole polynomial is 2. The terms in z squared y squared minus 25 have degree 4 and 0 so the degree of the whole polynomial is 4. The terms in 4 n cubed minus 8 n squared have degree 3 and 2 so the degree of the whole polynomial is 3. Trinomial examples: The terms in x squared minus 12 x plus 27 have degree 2, 1 and 0 so the degree of the whole polynomial is 2. The terms in 9 a squared plus 6 a b plus b squared have degree 2, 2, and 2 so the degree of the whole polynomial is 2. The terms in 6 m to the fourth power minus m cubed n squared plus 8 m n to the fifth power have degree 4, 5, and 6 so the degree of the whole polynomial is 6. The terms in z to the fourth power plus 3 z squared minus 1 have degree 4, 2, and 0 so the degree of the whole polynomial is 4. Polynomial examples: The terms in y minus 1 have degree 1 and 0 so the degree of the whole polynomial is 1. The terms in 3 y squared minus 2 y minus 5 have degree 2, 1, 0 so the degree of the whole polynomial is 2. The terms in 4 x to the fourth power plus x cubed plus eight x squared minus 9 x plus 1 have degree 4, 3, 2, 1, and 0 so the degree of the whole polynomial is 4.

Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form of a polynomial. Get in the habit of writing the term with the highest degree first.

Example 5.1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

ⓐ $7 y^{2} - 5 y + 3 7 y^{2} - 5 y + 3$ ⓑ $−2 a^{4} b^{2} −2 a^{4} b^{2}$ ⓒ $3 x^{5} - 4 x^{3} - 6 x^{2} + x - 8 3 x^{5} - 4 x^{3} - 6 x^{2} + x - 8$ ⓓ $2 y - 8 x y^{3} 2 y - 8 x y^{3}$ ⓔ 15

Show solution

Solution

	Polynomial	Number of terms	Type	Degree of terms	Degree of polynomial
ⓐ	$7 y^{2} - 5 y + 3 7 y^{2} - 5 y + 3$	3	Trinomial	2, 1, 0	2
ⓑ	$−2 a^{4} b^{2} −2 a^{4} b^{2}$	1	Monomial	4, 2	6
ⓒ	$3 x^{5} - 4 x^{3} - 6 x^{2} + x - 8 3 x^{5} - 4 x^{3} - 6 x^{2} + x - 8$	5	Polynomial	5, 3, 2, 1, 0	5
ⓓ	$2 y - 8 x y^{3} 2 y - 8 x y^{3}$	2	Binomial	1, 4	4
ⓔ	15	1	Monomial	0	0

Try It 5.1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

ⓐ $−5 −5$ ⓑ $8 y^{3} - 7 y^{2} - y - 3 8 y^{3} - 7 y^{2} - y - 3$ ⓒ $−3 x^{2} y - 5 x y + 9 x y^{3} −3 x^{2} y - 5 x y + 9 x y^{3}$ ⓓ $81 m^{2} - 4 n^{2} 81 m^{2} - 4 n^{2}$ ⓔ $−3 x^{6} y^{3} z −3 x^{6} y^{3} z$

Try It 5.2

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

ⓐ $64 k^{3} - 8 64 k^{3} - 8$ ⓑ $9 m^{3} + 4 m^{2} - 2 9 m^{3} + 4 m^{2} - 2$ ⓒ $\frac{5}{6} \frac{5}{6}$ ⓓ $8 a^{4} - 7 a^{3} b - 6 a^{2} b^{2} - 4 a b^{3} + 7 b^{4} 8 a^{4} - 7 a^{3} b - 6 a^{2} b^{2} - 4 a b^{3} + 7 b^{4}$ ⓔ $− p^{4} q^{3} − p^{4} q^{3}$

Add and Subtract Polynomials

We have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficients.

Example 5.2

Add or subtract: ⓐ $25 y^{2} + 15 y^{2} 25 y^{2} + 15 y^{2}$ ⓑ $16 p q^{3} - (−7 p q^{3}) . 16 p q^{3} - (−7 p q^{3}) .$

Show solution

Solution

ⓐ
$\begin{matrix} 25 y^{2} + 15 y^{2} \\ Combine like terms. & 40 y^{2} \end{matrix} \begin{matrix} Combine like terms. \end{matrix}$

ⓑ
$\begin{matrix} 16 p q^{3} - (−7 p q^{3}) \\ Combine like terms. & 23 p q^{3} \end{matrix} \begin{matrix} Combine like terms. \end{matrix}$

Try It 5.3

Add or subtract: ⓐ $12 q^{2} + 9 q^{2} 12 q^{2} + 9 q^{2}$ ⓑ $8 m n^{3} - (−5 m n^{3}) . 8 m n^{3} - (−5 m n^{3}) .$

Try It 5.4

Add or subtract: ⓐ $−15 c^{2} + 8 c^{2} −15 c^{2} + 8 c^{2}$ ⓑ $−15 y^{2} z^{3} - (−5 y^{2} z^{3}) . −15 y^{2} z^{3} - (−5 y^{2} z^{3}) .$

Remember that like terms must have the same variables with the same exponents.

Example 5.3

Simplify: ⓐ $a^{2} + 7 b^{2} - 6 a^{2} a^{2} + 7 b^{2} - 6 a^{2}$ ⓑ $u^{2} v + 5 u^{2} - 3 v^{2} . u^{2} v + 5 u^{2} - 3 v^{2} .$

Show solution

Solution

ⓐ
$\begin{matrix} a^{2} + 7 b^{2} - 6 a^{2} \\ Combine like terms. & −5 a^{2} + 7 b^{2} \end{matrix} \begin{matrix} Combine like terms. \end{matrix}$

ⓑ
$\begin{matrix} u^{2} v + 5 u^{2} - 3 v^{2} \\ \begin{matrix} There are no like terms to combine. \\ In this case, the polynomial is unchanged. \end{matrix} & u^{2} v + 5 u^{2} - 3 v^{2} \end{matrix} \begin{matrix} u^{2} v + 5 u^{2} - 3 v^{2} \\ \begin{matrix} There are no like terms to combine. \\ In this case, the polynomial is unchanged. \end{matrix} & u^{2} v + 5 u^{2} - 3 v^{2} \end{matrix}$

Try It 5.5

Add: ⓐ $8 y^{2} + 3 z^{2} - 3 y^{2} 8 y^{2} + 3 z^{2} - 3 y^{2}$ ⓑ $m^{2} n^{2} - 8 m^{2} + 4 n^{2} . m^{2} n^{2} - 8 m^{2} + 4 n^{2} .$

Try It 5.6

Add: ⓐ $3 m^{2} + n^{2} - 7 m^{2} 3 m^{2} + n^{2} - 7 m^{2}$ ⓑ $p q^{2} - 6 p - 5 q^{2} . p q^{2} - 6 p - 5 q^{2} .$

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

Example 5.4

Find the sum: $(7 y^{2} - 2 y + 9) + (4 y^{2} - 8 y - 7) . (7 y^{2} - 2 y + 9) + (4 y^{2} - 8 y - 7) .$

Show solution

Solution

$\begin{matrix} Identify like terms. & (\underset{____}{\underset{____}{7 y^{2}}} - \underset{___}{2 y} + 9) + (\underset{____}{\underset{____}{4 y^{2}}} - \underset{___}{8 y} - 7) \\ \begin{matrix} Rewrite without the parentheses, \\ rearranging to get the like terms together. \end{matrix} & \underset{_________}{\underset{_________}{7 y^{2} + 4 y^{2}}} - \underset{_______}{2 y - 8 y} + 9 - 7 \\ Combine like terms. & 11 y^{2} - 10 y + 2 \end{matrix} \begin{matrix} Identify like terms. & (\underset{____}{\underset{____}{7 y^{2}}} - \underset{___}{2 y} + 9) + (\underset{____}{\underset{____}{4 y^{2}}} - \underset{___}{8 y} - 7) \\ \begin{matrix} Rewrite without the parentheses, \\ rearranging to get the like terms together. \end{matrix} & \underset{_________}{\underset{_________}{7 y^{2} + 4 y^{2}}} - \underset{_______}{2 y - 8 y} + 9 - 7 \\ Combine like terms. & 11 y^{2} - 10 y + 2 \end{matrix}$

Try It 5.7

Find the sum: $(7 x^{2} - 4 x + 5) + (x^{2} - 7 x + 3) . (7 x^{2} - 4 x + 5) + (x^{2} - 7 x + 3) .$

Try It 5.8

Find the sum: $(14 y^{2} + 6 y - 4) + (3 y^{2} + 8 y + 5) . (14 y^{2} + 6 y - 4) + (3 y^{2} + 8 y + 5) .$

Be careful with the signs as you distribute while subtracting the polynomials in the next example.

Example 5.5

Find the difference: $(9 w^{2} - 7 w + 5) - (2 w^{2} - 4) . (9 w^{2} - 7 w + 5) - (2 w^{2} - 4) .$

Show solution

Solution

$\begin{matrix} (9 w^{2} - 7 w + 5) - (2 w^{2} - 4) \\ Distribute and identify like terms. & \underset{____}{\underset{____}{9 w^{2}}} - \underset{___}{7 w} + 5 - \underset{____}{\underset{____}{2 w^{2}}} + 4 \\ Rearrange the terms. & \underset{__________}{\underset{__________}{9 w^{2} - 2 w^{2}}} - \underset{___}{7 w} + 5 + 4 \\ Combine like terms. & 7 w^{2} - 7 w + 9 \end{matrix} \begin{matrix} (9 w^{2} - 7 w + 5) - (2 w^{2} - 4) \\ Distribute and identify like terms. & \underset{____}{\underset{____}{9 w^{2}}} - \underset{___}{7 w} + 5 - \underset{____}{\underset{____}{2 w^{2}}} + 4 \\ Rearrange the terms. & \underset{__________}{\underset{__________}{9 w^{2} - 2 w^{2}}} - \underset{___}{7 w} + 5 + 4 \\ Combine like terms. & 7 w^{2} - 7 w + 9 \end{matrix}$

Try It 5.9

Find the difference: $(8 x^{2} + 3 x - 19) - (7 x^{2} - 14) . (8 x^{2} + 3 x - 19) - (7 x^{2} - 14) .$

Try It 5.10

Find the difference: $(9 b^{2} - 5 b - 4) - (3 b^{2} - 5 b - 7) . (9 b^{2} - 5 b - 4) - (3 b^{2} - 5 b - 7) .$

To subtract $a a$ from $b, b,$ we write it as $b - a, b - a,$ placing the $b b$ first.

Example 5.6

Subtract $(p^{2} + 10 p q - 2 q^{2}) (p^{2} + 10 p q - 2 q^{2})$ from $(p^{2} + q^{2}) . (p^{2} + q^{2}) .$

Show solution

Solution

$\begin{matrix} (p^{2} + q^{2}) - (p^{2} + 10 p q - 2 q^{2}) \\ Distribute. & p^{2} + q^{2} - p^{2} - 10 p q + 2 q^{2} \\ Rearrange the terms, to put like terms together. & p^{2} - p^{2} - 10 p q + q^{2} + 2 q^{2} \\ Combine like terms. & −10 p q + 3 q^{2} \end{matrix} \begin{matrix} (p^{2} + q^{2}) - (p^{2} + 10 p q - 2 q^{2}) \\ Distribute. & p^{2} + q^{2} - p^{2} - 10 p q + 2 q^{2} \\ Rearrange the terms, to put like terms together. & p^{2} - p^{2} - 10 p q + q^{2} + 2 q^{2} \\ Combine like terms. & −10 p q + 3 q^{2} \end{matrix}$

Try It 5.11

Subtract $(a^{2} + 5 a b - 6 b^{2}) (a^{2} + 5 a b - 6 b^{2})$ from $(a^{2} + b^{2}) . (a^{2} + b^{2}) .$

Try It 5.12

Subtract $(m^{2} - 7 m n - 3 n^{2}) (m^{2} - 7 m n - 3 n^{2})$ from $(m^{2} + n^{2}) . (m^{2} + n^{2}) .$

Example 5.7

Find the sum: $(u^{2} - 6 u v + 5 v^{2}) + (3 u^{2} + 2 u v) . (u^{2} - 6 u v + 5 v^{2}) + (3 u^{2} + 2 u v) .$

Show solution

Solution

$\begin{matrix} (u^{2} - 6 u v + 5 v^{2}) + (3 u^{2} + 2 u v) \\ Distribute. & u^{2} - 6 u v + 5 v^{2} + 3 u^{2} + 2 u v \\ Rearrange the terms to put like terms together. & u^{2} + 3 u^{2} - 6 u v + 2 u v + 5 v^{2} \\ Combine like terms. & 4 u^{2} - 4 u v + 5 v^{2} \end{matrix} \begin{matrix} (u^{2} - 6 u v + 5 v^{2}) + (3 u^{2} + 2 u v) \\ Distribute. & u^{2} - 6 u v + 5 v^{2} + 3 u^{2} + 2 u v & Rearrange the terms to put like terms together. & u^{2} + 3 u^{2} - 6 u v + 2 u v + 5 v^{2} & Combine like terms. & 4 u^{2} - 4 u v + 5 v^{2} \end{matrix}$

Try It 5.13

Find the sum: $(3 x^{2} - 4 x y + 5 y^{2}) + (2 x^{2} - x y) . (3 x^{2} - 4 x y + 5 y^{2}) + (2 x^{2} - x y) .$

Try It 5.14

Find the sum: $(2 x^{2} - 3 x y - 2 y^{2}) + (5 x^{2} - 3 x y) . (2 x^{2} - 3 x y - 2 y^{2}) + (5 x^{2} - 3 x y) .$

When we add and subtract more than two polynomials, the process is the same.

Example 5.8

Simplify: $(a^{3} - a^{2} b) - (a b^{2} + b^{3}) + (a^{2} b + a b^{2}) . (a^{3} - a^{2} b) - (a b^{2} + b^{3}) + (a^{2} b + a b^{2}) .$

Show solution

Solution

$\begin{matrix} (a^{3} - a^{2} b) - (a b^{2} + b^{3}) + (a^{2} b + a b^{2}) \\ Distribute. & a^{3} - a^{2} b - a b^{2} - b^{3} + a^{2} b + a b^{2} \\ Rewrite without the parentheses, \\ rearranging to get the like terms together. & a^{3} - a^{2} b + a^{2} b - a b^{2} + a b^{2} - b^{3} \\ Combine like terms. & a^{3} - b^{3} \end{matrix} \begin{matrix} (a^{3} - a^{2} b) - (a b^{2} + b^{3}) + (a^{2} b + a b^{2}) \\ Distribute. & a^{3} - a^{2} b - a b^{2} - b^{3} + a^{2} b + a b^{2} & Rewrite without the parentheses, & rearranging to get the like terms together. & a^{3} - a^{2} b + a^{2} b - a b^{2} + a b^{2} - b^{3} & Combine like terms. & a^{3} - b^{3} \end{matrix}$

Try It 5.15

Simplify: $(x^{3} - x^{2} y) - (x y^{2} + y^{3}) + (x^{2} y + x y^{2}) . (x^{3} - x^{2} y) - (x y^{2} + y^{3}) + (x^{2} y + x y^{2}) .$

Try It 5.16

Simplify: $(p^{3} - p^{2} q) + (p q^{2} + q^{3}) - (p^{2} q + p q^{2}) . (p^{3} - p^{2} q) + (p q^{2} + q^{3}) - (p^{2} q + p q^{2}) .$

Evaluate a Polynomial Function for a Given Value

A polynomial function is a function defined by a polynomial. For example, $f (x) = x^{2} + 5 x + 6 f (x) = x^{2} + 5 x + 6$ and $g (x) = 3 x - 4 g (x) = 3 x - 4$ are polynomial functions, because $x^{2} + 5 x + 6 x^{2} + 5 x + 6$ and $3 x - 4 3 x - 4$ are polynomials.

Polynomial Function

A polynomial function is a function whose range values are defined by a polynomial.

In Graphs and Functions, where we first introduced functions, we learned that evaluating a function means to find the value of $f (x) f (x)$ for a given value of x. To evaluate a polynomial function, we will substitute the given value for the variable and then simplify using the order of operations.

Example 5.9

For the function $f (x) = 5 x^{2} - 8 x + 4 f (x) = 5 x^{2} - 8 x + 4$ find: ⓐ $f (4) f (4)$ ⓑ $f (−2) f (−2)$ ⓒ $f (0) . f (0) .$

Show solution

Solution

ⓐ



Simplify the exponents.
Multiply.
Simplify.

ⓑ



Simplify the exponents.
Multiply.
Simplify.

ⓒ



Simplify the exponents.
Multiply.
Simplify.

Try It 5.17

For the function $f (x) = 3 x^{2} + 2 x - 15, f (x) = 3 x^{2} + 2 x - 15,$ find ⓐ $f (3) f (3)$ ⓑ $f (−5) f (−5)$ ⓒ $f (0) . f (0) .$

Try It 5.18

For the function $g (x) = 5 x^{2} - x - 4, g (x) = 5 x^{2} - x - 4,$ find ⓐ $g (−2) g (−2)$ ⓑ $g (−1) g (−1)$ ⓒ $g (0) . g (0) .$

The polynomial functions similar to the one in the next example are used in many fields to determine the height of an object at some time after it is projected into the air. The polynomial in the next function is used specifically for dropping something from 250 ft.

Example 5.10

The polynomial function $h (t) = −16 t^{2} + 250 h (t) = −16 t^{2} + 250$ gives the height of a ball t seconds after it is dropped from a 250-foot tall building. Find the height after $t = 2 t = 2$ seconds.

Show solution

Solution

$\begin{matrix} h (t) = −16 t^{2} + 250 \\ To find h (2), substitute t = 2 . & h (2) = −16 {(2)}^{2} + 250 \\ Simplify. & h (2) = −16 \cdot 4 + 250 \\ Simplify. & h (2) = −64 + 250 \\ Simplify. & h (2) = 186 \\ After 2 seconds the height of the ball is 186 feet. \end{matrix} \begin{matrix} h (t) = −16 t^{2} + 250 \\ To find & h (2) = −16 {(2)}^{2} + 250 & Simplify. & h (2) = −16 \cdot 4 + 250 & Simplify. & h (2) = −64 + 250 & Simplify. & h (2) = 186 & After 2 seconds the height of the ball is 186 feet. \end{matrix}$

Try It 5.19

The polynomial function $h (t) = −16 t^{2} + 150 h (t) = −16 t^{2} + 150$ gives the height of a stone t seconds after it is dropped from a 150-foot tall cliff. Find the height after $t = 0 t = 0$ seconds (the initial height of the object).

Try It 5.20

The polynomial function $h (t) = −16 t^{2} + 175 h (t) = −16 t^{2} + 175$ gives the height of a ball t seconds after it is dropped from a 175-foot tall bridge. Find the height after $t = 3 t = 3$ seconds.

Add and Subtract Polynomial Functions

Just as polynomials can be added and subtracted, polynomial functions can also be added and subtracted.

Addition and Subtraction of Polynomial Functions

For functions $f (x) f (x)$ and $g (x), g (x),$

\begin{matrix} (f + g) (x) = f (x) + g (x) \\ (f - g) (x) = f (x) - g (x) \end{matrix} \begin{matrix} (f + g) (x) = f (x) + g (x) \\ (f - g) (x) = f (x) - g (x) \end{matrix}

Example 5.11

For functions $f (x) = 3 x^{2} - 5 x + 7 f (x) = 3 x^{2} - 5 x + 7$ and $g (x) = x^{2} - 4 x - 3, g (x) = x^{2} - 4 x - 3,$ find:

ⓐ $(f + g) (x) (f + g) (x)$ ⓑ $(f + g) (3) (f + g) (3)$ ⓒ $(f - g) (x) (f - g) (x)$ ⓓ $(f - g) (−2) . (f - g) (−2) .$

Show solution

Solution

ⓐ



Rewrite without the parentheses.
Put like terms together.
Combine like terms.

ⓑ In part (a) we found $(f + g) (x) (f + g) (x)$ and now are asked to find $(f + g) (3) . (f + g) (3) .$

$\begin{matrix} (f + g) (x) = 4 x^{2} - 9 x + 4 \\ To find (f + g) (3), substitute x = 3 . & (f + g) (3) = 4 {(3)}^{2} - 9 \cdot 3 + 4 \\ (f + g) (3) = 4 \cdot 9 - 9 \cdot 3 + 4 \\ (f + g) (3) = 36 - 27 + 4 \end{matrix} \begin{matrix} (f + g) (x) = 4 x^{2} - 9 x + 4 \\ To find & (f + g) (3) = 4 {(3)}^{2} - 9 \cdot 3 + 4 & (f + g) (3) = 4 \cdot 9 - 9 \cdot 3 + 4 & (f + g) (3) = 36 - 27 + 4 \end{matrix}$

Notice that we could have found $(f + g) (3) (f + g) (3)$ by first finding the values of $f (3) f (3)$ and $g (3) g (3)$ separately and then adding the results.

Find $f (3) . f (3) .$


Find $g (3) . g (3) .$


Find $(f + g) (3) . (f + g) (3) .$

ⓒ



Rewrite without the parentheses.
Put like terms together.
Combine like terms.

ⓓ

Try It 5.21

For functions $f (x) = 2 x^{2} - 4 x + 3 f (x) = 2 x^{2} - 4 x + 3$ and $g (x) = x^{2} - 2 x - 6, g (x) = x^{2} - 2 x - 6,$ find: ⓐ $(f + g) (x) (f + g) (x)$ ⓑ $(f + g) (3) (f + g) (3)$ ⓒ $(f - g) (x) (f - g) (x)$ ⓓ $(f - g) (−2) . (f - g) (−2) .$

Try It 5.22

For functions $f (x) = 5 x^{2} - 4 x - 1 f (x) = 5 x^{2} - 4 x - 1$ and $g (x) = x^{2} + 3 x + 8, g (x) = x^{2} + 3 x + 8,$ find ⓐ $(f + g) (x) (f + g) (x)$ ⓑ $(f + g) (3) (f + g) (3)$ ⓒ $(f - g) (x) (f - g) (x)$ ⓓ $(f - g) (−2) . (f - g) (−2) .$

Media

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Adding and Subtracting Polynomials

Practice Makes Perfect

Determine the Type of Polynomials

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial.

ⓐ $47 x^{5} - 17 x^{2} y^{3} + y^{2} 47 x^{5} - 17 x^{2} y^{3} + y^{2}$
ⓑ $5 c^{3} + 11 c^{2} - c - 8 5 c^{3} + 11 c^{2} - c - 8$
ⓒ $\frac{5}{9} a b + \frac{1}{3} b \frac{5}{9} a b + \frac{1}{3} b$
ⓓ 4
ⓔ $4 p q + 17 4 p q + 17$

ⓐ $x^{2} - y^{2} x^{2} - y^{2}$
ⓑ $−13 c^{4} −13 c^{4}$
ⓒ $a^{2} + 2 a b - 7 b^{2} a^{2} + 2 a b - 7 b^{2}$
ⓓ $4 x^{2} y^{2} - 3 x y + 8 4 x^{2} y^{2} - 3 x y + 8$
ⓔ 19

ⓐ $8 y - 5 x 8 y - 5 x$
ⓑ $y^{2} - 5 y z - 6 z^{2} y^{2} - 5 y z - 6 z^{2}$
ⓒ $y^{3} - 8 y^{2} + 2 y - 16 y^{3} - 8 y^{2} + 2 y - 16$
ⓓ $81 a b^{4} - 24 a^{2} b^{2} + 3 b 81 a b^{4} - 24 a^{2} b^{2} + 3 b$
ⓔ $−18 −18$

ⓐ $11 y^{2} 11 y^{2}$
ⓑ $−73 −73$
ⓒ $6 x^{2} - 3 x y + 4 x - 2 y + y^{2} 6 x^{2} - 3 x y + 4 x - 2 y + y^{2}$
ⓓ $4 y^{2} + 17 z^{2} 4 y^{2} + 17 z^{2}$
ⓔ $5 c^{3} + 11 c^{2} - c - 8 5 c^{3} + 11 c^{2} - c - 8$

ⓐ $5 a^{2} + 12 a b - 7 b^{2} 5 a^{2} + 12 a b - 7 b^{2}$
ⓑ $18 x y^{2} z 18 x y^{2} z$
ⓒ $5 x + 2 5 x + 2$
ⓓ $y^{3} - 8 y^{2} + 2 y - 16 y^{3} - 8 y^{2} + 2 y - 16$
ⓔ $−24 −24$

ⓐ $9 y^{3} - 10 y^{2} + 2 y - 6 9 y^{3} - 10 y^{2} + 2 y - 6$
ⓑ $−12 p^{3} q −12 p^{3} q$
ⓒ $a^{2} + 9 a b + 18 b^{2} a^{2} + 9 a b + 18 b^{2}$
ⓓ $20 x^{2} y^{2} - 10 a^{2} b^{2} + 30 20 x^{2} y^{2} - 10 a^{2} b^{2} + 30$
ⓔ 17

ⓐ $14 s - 29 t 14 s - 29 t$
ⓑ $z^{2} - 5 z - 6 z^{2} - 5 z - 6$
ⓒ $y^{3} - 8 y^{2} z + 2 y z^{2} - 16 z^{3} y^{3} - 8 y^{2} z + 2 y z^{2} - 16 z^{3}$
ⓓ $23 a b^{2} - 14 23 a b^{2} - 14$
ⓔ $−3 −3$

ⓐ $15 x y^{} 15 x y^{}$
ⓑ 15
ⓒ $6 x^{2} - 3 x y + 4 x - 2 y + y^{2} 6 x^{2} - 3 x y + 4 x - 2 y + y^{2}$
ⓓ $10 p - 9 q 10 p - 9 q$
ⓔ $m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1 m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1$

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

ⓐ ${7x}^{2} + 5 x^{2}^{2} + 5 x^{2}$
ⓑ $4a - 9 a$

10.

ⓐ ${4y}^{3} + 6 y^{3}^{3} + 6 y^{3}$
ⓑ $− y - 5 y − y - 5 y$

11.

ⓐ $−12 w + 18 w −12 w + 18 w$
ⓑ $7 x^{2} y - (−12 x^{2} y) 7 x^{2} y - (−12 x^{2} y)$

12.

ⓐ $−3 m + 9 m −3 m + 9 m$
ⓑ $15 y z^{2} - (−8 y z^{2}) 15 y z^{2} - (−8 y z^{2})$

13.

${7x}^{2} + 5 x^{2} + 4a - 9 a^{2} + 5 x^{2} +$

14.

${4y}^{3} + 6 y^{3} - y - 5 y^{3} + 6 y^{3} - y - 5 y$

15.

$−12 w + 18 w + 7 x^{2} y - (−12 x^{2} y) −12 w + 18 w + 7 x^{2} y - (−12 x^{2} y)$

16.

$−3 m + 9 m + 15 y z^{2} - (−8 y z^{2}) −3 m + 9 m + 15 y z^{2} - (−8 y z^{2})$

17.

ⓐ $−5 b - 17 b −5 b - 17 b$
ⓑ $3 x y - (−8 x y) + 5 x y 3 x y - (−8 x y) + 5 x y$

18.

ⓐ $−10 x - 35 x −10 x - 35 x$
ⓑ $17 m n^{2} - (−9 m n^{2}) + 3 m n^{2} 17 m n^{2} - (−9 m n^{2}) + 3 m n^{2}$

19.

ⓐ $12 a + 5 b - 22 a$
ⓑ $p q^{2} - 4 p - 3 q^{2} p q^{2} - 4 p - 3 q^{2}$

20.

ⓐ $14x - 3 y - 13 x$
ⓑ $a^{2} b - 4 a - 5 a b^{2} a^{2} b - 4 a - 5 a b^{2}$

21.

ⓐ $2 a^{2} + b^{2} - 6 a^{2} 2 a^{2} + b^{2} - 6 a^{2}$
ⓑ $x^{2} y - 3 x + 7 x y^{2} x^{2} y - 3 x + 7 x y^{2}$

22.

ⓐ $5 u^{2} + 4 v^{2} - 6 u^{2} 5 u^{2} + 4 v^{2} - 6 u^{2}$
ⓑ $12a + 8 b$

23.

ⓐ $x y^{2} - 5 x - 5 y^{2} x y^{2} - 5 x - 5 y^{2}$
ⓑ $19y + 5 z$

24.

$12 a + 5 b - 22 a + p q^{2} - 4 p - 3 q^{2} 12 a + 5 b - 22 a + p q^{2} - 4 p - 3 q^{2}$

25.

$14x - 3 y - 13 x + a^{2} b - 4 a - 5 a b^{2} 14x - 3 y - 13 x + a^{2} b - 4 a - 5 a b^{2}$

26.

$2 a^{2} + b^{2} - 6 a^{2} + x^{2} y - 3 x + 7 x y^{2} 2 a^{2} + b^{2} - 6 a^{2} + x^{2} y - 3 x + 7 x y^{2}$

27.

$5 u^{2} + 4 v^{2} - 6 u^{2} + 12a + 8 b 5 u^{2} + 4 v^{2} - 6 u^{2} +$

28.

$x y^{2} - 5 x - 5 y^{2} + 19y + 5 z x y^{2} - 5 x - 5 y^{2} +$

29.

Add: $4 a, −3 b, −8 a 4 a, −3 b, −8 a$

30.

Add: $4x, 3 y, −3 x$

31.

Subtract $5 x^{6} 5 x^{6}$ from $−12 x^{6} −12 x^{6}$

32.

Subtract $2 p^{4} 2 p^{4}$ from $−7 p^{4} −7 p^{4}$

In the following exercises, add the polynomials.

33.

$(5 y^{2} + 12 y + 4) + (6 y^{2} - 8 y + 7) (5 y^{2} + 12 y + 4) + (6 y^{2} - 8 y + 7)$

34.

$(4 y^{2} + 10 y + 3) + (8 y^{2} - 6 y + 5) (4 y^{2} + 10 y + 3) + (8 y^{2} - 6 y + 5)$

35.

$(x^{2} + 6 x + 8) + (−4 x^{2} + 11 x - 9) (x^{2} + 6 x + 8) + (−4 x^{2} + 11 x - 9)$

36.

$(y^{2} + 9 y + 4) + (−2 y^{2} - 5 y - 1) (y^{2} + 9 y + 4) + (−2 y^{2} - 5 y - 1)$

37.

$(8 x^{2} - 5 x + 2) + (3 x^{2} + 3) (8 x^{2} - 5 x + 2) + (3 x^{2} + 3)$

38.

$(7 x^{2} - 9 x + 2) + (6 x^{2} - 4) (7 x^{2} - 9 x + 2) + (6 x^{2} - 4)$

39.

$(5 a^{2} + 8) + (a^{2} - 4 a - 9) (5 a^{2} + 8) + (a^{2} - 4 a - 9)$

40.

$(p^{2} - 6 p - 18) + (2 p^{2} + 11) (p^{2} - 6 p - 18) + (2 p^{2} + 11)$

In the following exercises, subtract the polynomials.

41.

$(4 m^{2} - 6 m - 3) - (2 m^{2} + m - 7) (4 m^{2} - 6 m - 3) - (2 m^{2} + m - 7)$

42.

$(3 b^{2} - 4 b + 1) - (5 b^{2} - b - 2) (3 b^{2} - 4 b + 1) - (5 b^{2} - b - 2)$

43.

$(a^{2} + 8 a + 5) - (a^{2} - 3 a + 2) (a^{2} + 8 a + 5) - (a^{2} - 3 a + 2)$

44.

$(b^{2} - 7 b + 5) - (b^{2} - 2 b + 9) (b^{2} - 7 b + 5) - (b^{2} - 2 b + 9)$

45.

$(12 s^{2} - 15 s) - (s - 9) (12 s^{2} - 15 s) - (s - 9)$

46.

$(10 r^{2} - 20 r) - (r - 8) (10 r^{2} - 20 r) - (r - 8)$

In the following exercises, subtract the polynomials.

47.

Subtract $(9 x^{2} + 2) (9 x^{2} + 2)$ from $(12 x^{2} - x + 6) (12 x^{2} - x + 6)$

48.

Subtract $(5 y^{2} - y + 12) (5 y^{2} - y + 12)$ from $(10 y^{2} - 8 y - 20) (10 y^{2} - 8 y - 20)$

49.

Subtract $(7 w^{2} - 4 w + 2) (7 w^{2} - 4 w + 2)$ from $(8 w^{2} - w + 6) (8 w^{2} - w + 6)$

50.

Subtract $(5 x^{2} - x + 12) (5 x^{2} - x + 12)$ from $(9 x^{2} - 6 x - 20) (9 x^{2} - 6 x - 20)$

In the following exercises, find the difference of the polynomials.

51.

Find the difference of $(w^{2} + w - 42) (w^{2} + w - 42)$ and $(w^{2} - 10 w + 24) (w^{2} - 10 w + 24)$

52.

Find the difference of $(z^{2} - 3 z - 18) (z^{2} - 3 z - 18)$ and $(z^{2} + 5 z - 20) (z^{2} + 5 z - 20)$

In the following exercises, add the polynomials.

53.

$(7 x^{2} - 2 x y + 6 y^{2}) + (3 x^{2} - 5 x y) (7 x^{2} - 2 x y + 6 y^{2}) + (3 x^{2} - 5 x y)$

54.

$(−5 x^{2} - 4 x y - 3 y^{2}) + (2 x^{2} - 7 x y) (−5 x^{2} - 4 x y - 3 y^{2}) + (2 x^{2} - 7 x y)$

55.

$(7 m^{2} + m n - 8 n^{2}) + (3 m^{2} + 2 m n) (7 m^{2} + m n - 8 n^{2}) + (3 m^{2} + 2 m n)$

56.

$(2 r^{2} - 3 r s - 2 s^{2}) + (5 r^{2} - 3 r s) (2 r^{2} - 3 r s - 2 s^{2}) + (5 r^{2} - 3 r s)$

In the following exercises, add or subtract the polynomials.

57.

$(a^{2} - b^{2}) - (a^{2} + 3 a b - 4 b^{2}) (a^{2} - b^{2}) - (a^{2} + 3 a b - 4 b^{2})$

58.

$(m^{2} + 2 n^{2}) - (m^{2} - 8 m n - n^{2}) (m^{2} + 2 n^{2}) - (m^{2} - 8 m n - n^{2})$

59.

$(p^{3} - 3 p^{2} q) + (2 p q^{2} + 4 q^{3}) - (3 p^{2} q + p q^{2}) (p^{3} - 3 p^{2} q) + (2 p q^{2} + 4 q^{3}) - (3 p^{2} q + p q^{2})$

60.

$(a^{3} - 2 a^{2} b) + (a b^{2} + b^{3}) - (3 a^{2} b + 4 a b^{2}) (a^{3} - 2 a^{2} b) + (a b^{2} + b^{3}) - (3 a^{2} b + 4 a b^{2})$

61.

$(x^{3} - x^{2} y) - (4 x y^{2} - y^{3}) + (3 x^{2} y - x y^{2}) (x^{3} - x^{2} y) - (4 x y^{2} - y^{3}) + (3 x^{2} y - x y^{2})$

62.

$(x^{3} - 2 x^{2} y) - (x y^{2} - 3 y^{3}) - (x^{2} y - 4 x y^{2}) (x^{3} - 2 x^{2} y) - (x y^{2} - 3 y^{3}) - (x^{2} y - 4 x y^{2})$

Evaluate a Polynomial Function for a Given Value

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

63.

For the function $f (x) = 8 x^{2} - 3 x + 2, f (x) = 8 x^{2} - 3 x + 2,$ find:
ⓐ $f (5) f (5)$ ⓑ $f (−2) f (−2)$ ⓒ $f (0) f (0)$

64.

For the function $f (x) = 5 x^{2} - x - 7, f (x) = 5 x^{2} - x - 7,$ find:
ⓐ $f (−4) f (−4)$ ⓑ $f (1) f (1)$ ⓒ $f (0) f (0)$

65.

66.

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

67.

A painter drops a brush from a platform 75 feet high. The polynomial function $h (t) = −16 t^{2} + 75 h (t) = −16 t^{2} + 75$ gives the height of the brush t seconds after it was dropped. Find the height after $t = 2 t = 2$ seconds.

68.

A girl drops a ball off the cliff into the ocean. The polynomial $h (t) = −16 t^{2} + 200 h (t) = −16 t^{2} + 200$ gives the height of a ball t seconds after it is dropped. Find the height after $t = 3 t = 3$ seconds.

69.

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial function $R (p) = −4 p^{2} + 420 p . R (p) = −4 p^{2} + 420 p .$ Find the revenue received when $p = 60 p = 60$ dollars.

70.

A manufacturer of the latest basketball shoes has found that the revenue received from selling the shoes at a cost of p dollars each is given by the polynomial $R (p) = −4 p^{2} + 420 p . R (p) = −4 p^{2} + 420 p .$ Find the revenue received when $p = 90 p = 90$ dollars.

71.

The polynomial $C (x) = 6 x^{2} + 90 x C (x) = 6 x^{2} + 90 x$ gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and height 6 feet. Find the cost of producing a box with $x = 4 x = 4$ feet.

72.

The polynomial $C (x) = 6 x^{2} + 90 x C (x) = 6 x^{2} + 90 x$ gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and height 4 feet. Find the cost of producing a box with $x = 6 x = 6$ feet.

Add and Subtract Polynomial Functions

73.

$f (x) = 2 x^{2} - 4 x + 1 f (x) = 2 x^{2} - 4 x + 1$ and $g (x) = 5 x^{2} + 8 x + 3 g (x) = 5 x^{2} + 8 x + 3$

74.

$f (x) = 4 x^{2} - 7 x + 3 f (x) = 4 x^{2} - 7 x + 3$ and $g (x) = 4 x^{2} + 2 x - 1 g (x) = 4 x^{2} + 2 x - 1$

75.

$f (x) = 3 x^{3} - x^{2} - 2 x + 3 f (x) = 3 x^{3} - x^{2} - 2 x + 3$ and $g (x) = 3 x^{3} - 7 x g (x) = 3 x^{3} - 7 x$

76.

$f (x) = 5 x^{3} - x^{2} + 3 x + 4 f (x) = 5 x^{3} - x^{2} + 3 x + 4$ and $g (x) = 8 x^{3} - 1 g (x) = 8 x^{3} - 1$

Writing Exercises

77.

Using your own words, explain the difference between a monomial, a binomial, and a trinomial.

78.

Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of 5.

79.

Ariana thinks the sum $6 y^{2} + 5 y^{4} 6 y^{2} + 5 y^{4}$ is $11 y^{6} . 11 y^{6} .$ What is wrong with her reasoning?

80.

Is every trinomial a second degree polynomial? If not, give an example.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

The figure shows a table with six rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is "confidently", the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “identify polynomials, monomials, binomials, and trinomials”, “determine the degree of polynomials”, “add and subtract monomials”, “add and subtract polynomials”, and “evaluate a polynomial for a given value”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.