6-3 POLYNOMIALSChapter 6

OBJECTIVES

Classify polynomials and write polynomials in standard form.

Evaluate polynomial expressions.

MONOMIALS

A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents.

The degree of a monomial is the sum of the exponents of the variables. A constant has degree 0.

EXAMPLE 1: FINDING THE DEGREE OF A MONOMIAL

Find the degree of each monomial. A. 4p4q3

Sol: The degree is 7 Add the exponents of the variables: 4 + 3 = 7.

B. 7ed The degree is 2. Add the exponents of the

variables: 1+ 1 = 2. C. 3 Sol:The degree is 0.

CHECK IT OUT ! EXAMPLES

Find the degree of each monomial. A. 1.5k2m B. 4x C. 2c3

D.3yxz

POLYNOMIALS

A polynomial is a monomial or a sum or difference of monomials.

The degree of a polynomial is the degree of the term with the greatest degree.

EXAMPLE 2: FINDING THE DEGREE OF A POLYNOMIAL

Find the degree of each polynomial. A. 11x7 + 3x3

Sol:

11x7: degree 7 3x3: degree 3 The degree of the polynomial is the greatest

degree, 7.

The degree of the polynomial is the greatest degree, 4.

B.

CHECK IT OUT ! EXAMPLE

A. x3y2 + x2y3 – x4 + 2 Sol: The degree of the polynomial is the

greatest degree, 5.

STANDARD FORM OF A POLYNOMIAL

The terms of a polynomial may be written in any order. However, polynomials that contain only one variable are usually written in standard form.

The standard form of a polynomial that contains one variable is written with the terms in order from greatest degree to least degree. When written in standard form, the coefficient of the first term is called the leading coefficient.

EXAMPLE 3A: WRITING POLYNOMIALS IN STANDARD FORM

Write the polynomial in standard form. Then give the leading coefficient.

6x – 7x5 + 4x2 + 9 Sol: Find the degree of each term. Then arrange

them in descending order:

6x – 7x5 + 4x2 + 9 –7x5 + 4x2 + 6x + 9

Degree 1 5 2 0 5 2 1 0–7x5 + 4x2 + 6x + 9.The standard form is The leading

coefficient is –7.

EXAMPLE 3B: WRITING POLYNOMIALS IN STANDARD FORM

Write the polynomial in standard form. Then give the leading coefficient.

y2 + y6 – 3y Sol: Find the degree of each term. Then arrange

them in descending order:y2 + y6 – 3y y6 + y2 – 3y

Degree 2 6 1 2 16The standard form is The leading coefficient is 1.

y6 + y2 – 3y.

CHECK IT OUT! EXAMPLE

Write the polynomial in standard form. Then give the leading coefficient.

16 – 4x2 + x5 + 9x3

POLYNOMIALS

Some polynomials have special names based on their degree and the number of terms they have. Degree Name

0

1

2

Constant

Linear

Quadratic

3

4

5

6 or more 6th,7th,degree and so on

Cubic

Quartic

Quintic

NameTerms

Monomial

Binomial

Trinomial

Polynomial4 or more

1

2

3

EXAMPLE 4: CLASSIFYING POLYNOMIALS

Classify each polynomial according to its degree and number of terms.

A. 5n3 + 4n Degree 3 Terms 2 5n3 + 4n is a cubic binomial. B. 4y6 – 5y3 + 2y – 9 Degree 6 Terms 4

C. –2x Degree 1 Terms 1 –2x is a linear monomial.

4y6 – 5y3 + 2y – 9 is a

6th-degree polynomial.

CHECK IT OUT!

Classify each polynomial according to its degree and number of terms.

b. 6 c. –3y8 + 18y5 + 14y

a. x3 + x2 – x + 2

APPLICATION

A tourist accidentally drops her lip balm off the Golden Gate Bridge. The bridge is 220 feet from the water of the bay. The height of the lip balm is given by the polynomial –16t2 + 220, where t is time in seconds. How far above the water will the lip balm be after 3 seconds?

Sol: After 3 seconds the lip balm will be 76 feet

from the water.

STUDENT GUIDED PRACTICE

DO even problems from 4-21 in your book page 409

HOMEWORK

Do even problems 27-42 in your book page 409

CLOSURE

Today we learned about polynomials Next class we are going o learn about

adding and subtracting polynomials