Introduction to Polynomials Learning Targets Identifying Parts Of A Monomial I will be able to: Classify polynomials by the number of terms Classify Polynomials.

Introduction to Polynomials

Learning Targets

Identifying Parts Of A Monomial

I will be able to:

Classify polynomials by the number of terms

Classify Polynomials By Degree

IDENTIFYING PARTS OF A MONOMIAL

Coefficient

Variable

Exponent

Let’s try an example: Identify the coefficient, variable, and exponent:

Coefficient

Variable

Exponent

WAYS TO CLASSIFY POLYNOMIALSWe can classify polynomials by the number of terms:Monomial: 1 term Think about other words with the prefix

mono: monotone, monochromatic, monologue

Binomial: 2 terms Think about other words with the prefix bi: bicycle, bifocals, bimonthly

Trinomial: 3 terms Think about other words with the prefix tri: tricycle, triathlon, triceratops

Polynomial: 4 or more terms Think about other words with the prefix poly: polytheistic, polygon

Let’s take a closer look at classifying polynomials by number of terms...

Polynomials are fun!

CLASSIFYING POLYNOMIALS BY NUMBER OF TERMSMonomial: a number, a variable, or the product of a number and one or more variables. We are also going to call this a term.

Let’s check out some examples of monomials:

A monomial with no variables is called a constant.

CLASSIFYING POLYNOMIALS BY NUMBER OF TERMS

Binomial: a polynomial with 2 terms

Let’s check out some examples of binomials:

Trinomial: a polynomial with 3 terms

Let’s check out some examples of trinomials:

CLASSIFYING POLYNOMIALS BY DEGREE

Finding the degree of a Monomial: The sum of the exponents of its variables.

Example 1:

Finding the degree of a Polynomial: The same as that of its term with the greatest degree.

Example 1:

Example 2:

Example 2:

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

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

B. 7ed

C. 3

Check It Out! Example 1

Find the degree of each monomial.

a. 1.5k2m

b. 4x

b. 2c3

CLASSIFYING POLYNOMIALS BY DEGREE

Finding the degree of a Polynomial: The same as that of its term with the greatest degree.

Example 1:

Example 2:

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

Find the degree of each polynomial.

Example 2: Finding the Degree of a Polynomial

And its name

A. 11x7 + 3x3

11x7: degree 7 3x3: degree 3

The degree of the polynomial is the greatest degree, 7, so it’s 7th.

Find the degree of each term.

B.

The degree of the polynomial is the greatest degree, 4, so it’s quartic.

Check It Out! Example 2

Find the degree and the name of each polynomial.

a. 5x – 6

b. x3y2 + x2y3 – x4 + 2

CLASSIFYING POLYNOMIALS BY DEGREE

Degree Name Example

NON-EXAMPLES OF POLYNOMIALS

Fractions, Division

Square Roots

Variables as the exponent

Negatives as the exponent

Remember...these are NOT polynomials!

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.

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

Example 3A: Writing Polynomials in Standard Form

6x – 7x5 + 4x2 + 9

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.

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

Example 3B: Writing Polynomials in Standard Form

y2 + y6 − 3y

Check It Out! Example 3a

Write the polynomial in standard form. Give the leading coefficient. Then name it by degree and number of terms.

16 – 4x2 + x5 + 9x3

18y5 – 3y8 + 14y

Check It Out! Example 3b

Write the polynomial in standard form. Give the leading coefficient. Then name it by degree and number of terms.

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

Example 4: Classifying Polynomials

A. 5n3 + 4nDegree 3 Terms 2

5n3 + 4n is a cubic binomial.

B. 4y6 – 5y3 + 2y – 9

C. –2x

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

D. x3 + x2 – x + 2

E. 6

F. –3y8 + 18y5 + 14y

Lesson Closing

Find the degree of each polynomial.

1. 7a3b2 – 2a4 + 4b – 15

2. 25x2 – 3x4

Write each polynomial in standard form. Then

give the leading coefficient.

3. 24g3 + 10 + 7g5 – g2

4. 14 – x4 + 3x2

4

5

–x4 + 3x2 + 14; –1

7g5 + 24g3 – g2 + 10; 7

Lesson Closing: Part II

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

5. 18x2 – 12x + 5 quadratic trinomial

6. 2x4 – 1 quartic binomial