Chapter 5: Polynomials Chapter 5: Polynomials and Polynomial Functionsand Polynomial Functions5.1: Polynomial Functions

DefinitionsDefinitionsA monomial is a real number, a

variable, or a product of a real number and one or more variables with whole number exponents.◦Examples:

The degree of a monomial in one variable is the exponent of the variable.

2 35, , 3 , 4x wy x

DefinitionsDefinitionsA polynomial is a monomial or a

sum of monomials.◦Example:

The degree of a polynomial in one variable is the greatest degree among its monomial terms.◦Example:

23 2 5xy x

24 7x x

DefinitionsDefinitionsA polynomial function is a

polynomial of the variable x.◦A polynomial function has

distinguishing “behaviors” The algebraic form tells us about the

graph The graph tells us about the algebraic

form

DefinitionsDefinitionsThe standard form of a polynomial function arranges the terms by degree in descending order◦Example:

3 2( ) 4 3 5 2P x x x x

DefinitionsDefinitionsPolynomials are classified by

degree and number of terms.◦Polynomials of degrees zero through

five have specific names and polynomials with one through three terms also have specific names.

Degree

Name

0 Constant

1 Linear

2 Quadratic

3 Cubic

4 Quartic

5 Quintic

Number ofTerms

Name

1 Monomial

2 Binomial

3 Trinomial

4+ Polynomial with ___ terms

ExampleExampleWrite each polynomial in standard form. Then classify it by degree and by number of terms.

23 9 5x x

ExampleExampleWrite each polynomial in standard form. Then classify it by degree and by number of terms.

5 23 4 2 10x x

Polynomial BehaviorPolynomial BehaviorThe degree of a polynomial

function ◦Affects the shape of its graph◦Determines the number of turning points (places where the graph changes direction)

◦Affects the end behavior (the directions of the graph to the far left and to the far right)

Polynomial BehaviorPolynomial BehaviorThe graph of a polynomial function

of degree n has at most n – 1 turning points.◦Odd Degree = even number of turning

points◦Even Degree = odd number of turning

pointsThink about this:

◦If a polynomial has degree 2, how many turning points can it have?

◦If a polynomial has degree 3, how many turning points can it have?

Polynomial BehaviorPolynomial Behavior

End behavior is determined by the leading term nax

Polynomial Behavior Polynomial Behavior ExamplesExamples

4 34 6y x x x

2 2y x x

3y x

3 2y x x

ExampleExampleDetermine the end behavior of the graph of each polynomial function.34 3y x x

ExampleExampleDetermine the end behavior of the graph of each polynomial function.4 3 22 8 8 2y x x x

Increasing and DecreasingIncreasing and DecreasingRemember: We read from left to

right!

A function is increasing when the y-values increase as the x-values increase

A function is decreasing when the y-values decrease as the x-values increase

Example: Identify the parts Example: Identify the parts of the graph that are of the graph that are increasing or decreasingincreasing or decreasing

Example: Identify the parts Example: Identify the parts of the graph that are of the graph that are increasing or decreasingincreasing or decreasing

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