CHAPTER 8 Polynomials

# 8.1 Add and Subtract Polynomials

Learning Objectives

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

• Identify polynomials, monomials, binomials, and trinomials
• Determine the degree of polynomials
• Evaluate a polynomial for a given value

# Identify Polynomials, Monomials, Binomials and Trinomials

You have learned that a term is a constant or the product of a constant and one or more variables. When it is of the form , where is a constant and is a whole number, it is called a monomial. Some examples of monomial are , and .

Monomials

A monomial is a term of the form , where is a constant and is a positive 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 monomials 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.

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.

EXAMPLE 1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial.

Solution
 Polynomial Number of terms Type a) Trinomial b) Monomial c) Polynomial d) Binomial e) Monomial

TRY IT 1.1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

a) b) c) d) e)

a) monomial b) polynomial c) trinomial d) binomial e) monomial

TRY IT 1.2

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

a) b) c) d) e)

a) binomial b) trinomial c) monomial d) polynomial e) monomial

# Determine the Degree of 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—it has no variable.

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.

A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees. Get in the habit of writing the term with the highest degree first.

EXAMPLE 2

Find the degree of the following polynomials.

Solution
 a) The exponent of is one. The degree is 1. b) The highest degree of all the terms is 3. The degree is 3. c) The degree of a constant is 0. The degree is 0. d) The highest degree of all the terms is 2. The degree is 2. e) The highest degree of all the terms is 3. The degree is 3.

EXAMPLE 2.1

Find the degree of the following polynomials:

a) b) c) d) e)

a) b) c) d) 3 e) 0

TRY IT 2.2

Find the degree of the following polynomials:

a) b) c) d) e)

a) b) c) d) 2 e) 3

You 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 coefficient.

EXAMPLE 3

Solution
 Combine like terms.

TRY IT 3.1

TRY 3.2

EXAMPLE 4

Subtract: .

Solution
 Combine like terms.

TRY IT 4.1

Subtract: .

TRY IT 4.2

Subtract: .

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

EXAMPLE 5

Simplify: .

Solution
 Combine like terms.

TRY IT 5.1

TRY IT 5.2

EXAMPLE 6

Simplify: .

Solution
 There are no like terms to combine.

TRY IT 6.1

Simplify: .

There are no like terms to combine.

TRY IT 6.2

Simplify: .

There are no like terms to combine.

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 7

Find the sum: .

Solution
 Identify like terms. Rearrange to get the like terms together. Combine like terms.

TRY IT 7.1

Find the sum: .

TRY IT 7.2

Find the sum: .

EXAMPLE 8

Find the difference: .

Solution
 Distribute and identify like terms. Rearrange the terms. Combine like terms.

TRY IT 8.1

Find the difference: .

TRY IT 8.2

Find the difference: .

EXAMPLE 9

Subtract: from .

Solution
 Distribute and identify like terms. Rearrange the terms. Combine like terms.

TRY IT 9.1

Subtract: from .

TRY IT 9.2

Subtract: from .

EXAMPLE 10

Find the sum: .

Solution
 Distribute. Rearrange the terms, to put like terms together. Combine like terms.

EXAMPLE 10.1

Find the sum: .

EXAMPLE 10.2

Find the sum: .

EXAMPLE 11.1

Find the difference: .

Solution
 Distribute. Rearrange the terms, to put like terms together. Combine like terms.

TRY IT 11.1

Find the difference: .

TRY IT 11.2

Find the difference: .

EXAMPLE 12

Simplify: .

Solution
 Distribute. Rearrange the terms, to put like terms together. Combine like terms.

TRY IT 12.1

Simplify: .

TRY IT 12.2

Simplify: .

# Evaluate a Polynomial for a Given Value

We have already learned how to evaluate expressions. Since polynomials are expressions, we’ll follow the same procedures to evaluate a polynomial. We will substitute the given value for the variable and then simplify using the order of operations.

EXAMPLE 13

Evaluate when

Solution
 a) Simplify the exponents. Multiply. Simplify.
 b) Simplify the exponents. Multiply. Simplify.
 c) Simplify the exponents. Multiply. Simplify.

TRY IT 13.1

Evaluate: when

a) b) c)

TRY IT 13.2

Evaluate: when

a) b) c)

EXAMPLE 14

The polynomial gives the height of a ball seconds after it is dropped from a 250 foot tall building. Find the height after seconds.

Solution
 Substitute . Simplify. Simplify. Simplify. After 2 seconds the height of the ball is 186 feet.

TRY IT 14.1

The polynomial gives the height of a ball seconds after it is dropped from a 250-foot tall building. Find the height after seconds.

TRY IT 14.2

The polynomial gives the height of a ball seconds after it is dropped from a 250-foot tall building. Find the height after seconds.

EXAMPLE 15

The polynomial gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with feet and feet.

Solution
 Simplify. Simplify. Simplify. The cost of producing the box is $456. TRY IT 15.1 The polynomial gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with feet and feet. Show answer$576

TRY IT 15.2

The polynomial gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with feet and feet.