Polynomial FunctionsPolynomial FunctionsA function defined by an equation in the formA function defined by an equation in the form

1 21 2 1 0

n n nn n ny a x a x a x a x a

where is a non-negative integer and the

are constants.

'na sn

Graphs of Polynomial Graphs of Polynomial FunctionsFunctions

ContinuousContinuous

SmoooooooothSmooooooooth

Leading Coefficient TestLeading Coefficient Test

Real Zeros of the functionReal Zeros of the function

x

y

x

y

Linear

Quadratic

Cubic

x

y

General Shapes of FunctionsGeneral Shapes of Functions

Quartic Quintic

General Shapes of FunctionsGeneral Shapes of Functions

x

y

x

y

General Shapes of FunctionsGeneral Shapes of Functions

x

y

Linear

x

y

Cubic

Quintic

x

y

General Shapes of FunctionsGeneral Shapes of Functions

x

y

Quadratic Quartic

x

y

Leading Coefficient Test

Remember, when we talk about increasing or decreasing, rising or falling, we always are going from left to right!

Describes end behavior

Positive leading coefficients always end up rising

Negative leading coefficients always end up falling

Odd degrees start and end in opposite directionsEven degrees start and end in the same direction

Real Zeros of Polynomial FunctionsReal Zeros of Polynomial Functions

If If f f is a polynomial function andis a polynomial function and aa is a is a real number, then the following real number, then the following statements are equivalent:statements are equivalent:

x=ax=a is a is a zerozero of the function f of the function f x=ax=a is a is a solutionsolution of the polynomial of the polynomial

f(x)=0f(x)=0 (x-a) (x-a) is a is a factorfactor of of f f (a,0) (a,0) is anis an xx-intercept-intercept of the graph of the graph

ofof f(x) f(x)

Find all real zeros of

3 2 2f x x x x

Solution:

3 2 2f x x x x

3 20 2x x x 20 2x x x

0 1 2x x x

0 1 2x x x

0 1 0 2 0x or x or x

0, 1,2x

Find the real zeros of

5 3 24 3f x x x x

Use your calculator to graph and find the zeros!

x

y

IT’S UGLY!

Find the real zeros of

4 2f x x x

Factor to find the zeros!

1 2 3 4 5 6 7-1-2-3-4-5-6-7

1

2

3

4

5

-1

-2

-3

-4

-5

x

yNote:This function “bounces” off the x-axis at x=0. This means that there is a double root there

f

1 2 3-1-2-3

1

2

3

4

5

-1

x

y

4f x x 1 2 3-1-2-3

1

2

3

4

5

-1

x

y

8f x x

The more roots at a particular spot …

The flatter the graph becomes there