Chapter 9 Section 1

Polynomials

Vocabulary

Polynomial: The sum of one or more monomials is called a polynomial.

Monomial: A monomial is a number, a variable, or a product of numbers and variables that have only positive exponents.

Binomial: A polynomial with two terms is a binomial.

Trinomial: A polynomial with three terms is a trinomial.

What You’ll Learn

You’ll learn to identify and classify polynomials and find their degree.

Why it is important

One example of why it is important is medicine. Doctors can use

polynomials to study the heart.

A cube is a solid figure in which all the faces are square. Suppose you wanted to paint the cube shown below. You would need to find the surface area of the cube to determine how much paint to buy.

The area of each face of the cube is x ∙ x or x2. There are six faces to

paint. x2 + x2 + x2 + x2 + x2 + x2 = 6x2

So, the surface area of the cube is 6x2 square feet.

x ft.

x ft.x ft.

The expression 6x2 is called a monomial. A monomial is a number, a variable, or a product of numbers

and variables that have only positive exponents. A monomial cannot have

a variable as an exponent. Monomials Not Monomials

-4 A number 2x Has a variable as an exponent

Y A variable x2 + 3 Includes addition

a2 The product of variables

5a-2 Includes a negative exponent

½ x2y The product of numbers and variables

3x

Includes division

Example 1

Determine whether each expression is a monomial. Explain why or

why not.

-6ab

-6ab is a monomial. It is the product of a number and

variables.

Example 2

Determine whether each expression is a monomial. Explain why or why not.

m2 - 4

m2 – 4 is not a monomial, because it includes subtraction.

Your Turn

Determine whether each expression is a monomial. Explain why or why not.

10

10 is a monomial, because it is a number.

Your Turn

Determine whether each expression is a monomial. Explain why or why not.

5z-3

5z-3 is not a monomial, because it includes a negative exponent.

Your Turn

Determine whether each expression is a monomial. Explain why or why

not.

6x

This is not a monomial, because it includes division.

Your Turn

Determine whether each expression is a monomial. Explain why or why not.

x2

x2 is a monomial, because it is a product of variables.

The sum of one or more monomials is called a polynomial. For example,

x3 + x2 + x + 2is a polynomial. The terms of the polynomial are x3, x2, x, and 2.

Special names are given to polynomials with two or three terms. A polynomial with two

terms is a binomial. A polynomial with three terms is a

trinomial. Here are some examples. Binomial Trinomial

x + 2 a + b + c

5c - 4 x2 + 5x - 7

4w2 - w 3a2 + 5ab + 2b2

Example 3State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

2m – 7The expression 2m – 7 can be written as 2m + (-7). So, it is a

polynomial. Since it can be written as the sum of two

monomials, 2m and -7, it is a binomial.

Example 4State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

x2 + 3x – 4 - 5The expression x2 + 3x – 4 – 5 can

be written as x2 + 3x + (-9).So, it is a polynomial. Since it can

be written as the sum of three monomials, it is a trinomial.

Example 5State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

5 - 3 2x

The expression is not a polynomial since it is not a monomial. It

contains division.

Your TurnState whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

5a – 9 + 3Yes, Binomial

Your TurnState whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

4m-2 + 2No, Cannot have a negative

exponent

Your TurnState whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

3y2 – 6 + 7yYes, Trinomial

The terms of a polynomial are usually arranged so that the powers of one variable are in

descending or ascending order. Polynomial Descending Order Ascending Order

2x + x2 + 1 x2 + 2x + 1 1 + 2x + x2

3y2 + 5y3 + y 5y3 + 3y2 + y y + 3y2 + 5y3

x2 + y2 + 3xy x2 + 3xy + y2 y2 + 3xy + x2

2xy + y2 + x2 y2 + 2xy + x2 x2 + 2xy + y2

Degree

The degree of a monomial is the sum of the exponents of the variables. Monomial Degree

-3x2 2

5pq2 1 + 2 = 3

2 0

To find the degree of a polynomial, you must find the degree of each term. The degree of the polynomial is the greatest of the degrees of its term. Polynomial Terms Degree of the

TermsDegree of the

Polynomial

2n + 7 2n, 7 1, 0 1

3x2 + 5x 3x2, 5x 2, 1 2

a6 + 2a3 + 1 a6, 2a3, 1 6, 3, 0 6

5x4 – 4a2b6 + 3x

5x4, 4a2b6, 3x 4, 8, 1 8

Example 6

Find the degree of each polynomial.

5a2 + 3

So, the degree of 5a2 + 3 is 2.

Term Degree

5a2 2

3 0

Example 6

Find the degree of each polynomial.

6x2 – 4x2y – 3xy

So, the degree of 6x2 – 4x2y – 3xyis 3.

Term Degree

6x2 2

4x2y 2 + 1 or 3

3xy 1 + 1 or 2

Your Turn

Find the degree of each polynomial.

3x2 – 7x

2

Your Turn

Find the degree of each polynomial.

8m3 – 2m2n2 + 5

4