**Polynomial Functions**

**Polynomial in One Variable**

A polynomial function is a function of the form

Find Degree and

Leading Coefficients

Polynomial Function

It represents the function of a polynomial equation.

Functional Values

of Variables

Graphs of Polynomial Functions

End Behavior of a Polynomial Function

Graphs of Polynomial Functions

Example: Polynomial in one variable

It contains only one variable, x.

The degree of the polynomial is the degree of the leading term.

The a are real numbers and are called coefficients.

n

Yes

The degree is 4 and the leading coefficient is 6.

No

It contains two variables, x and y.

No

The term 2/x cannot be written in the form

x , where n is a nonnegative integer.

n

Yes

Rewrite the expression according to the degree of exponents,

Examples:

quadratic polynomial

function

cubic polynomial

function

Evaluate a

Polynomial Function

Replace x with 3.

Simplify.

Replace x with -3.

Simplify.

Original function

Replace x with y .

2

Property of powers

To evaluate h(k + 1), replace y in h(y) with k + 1.

To evaluate 2h(y), replace y with k in h(y), then multiply the expression by 2.

Now evaluate h(k + 1) - 2h(k).

Constant function

Degree 0

Linear function

Degree 1

Quadratic function

Degree 2

Cubic function

Degree 3

Quartic function

Degree 4

Quintic function

Degree 5

Degree: even

Leading Coefficient: positive

End Behavior:

Degree: odd

Leading Coefficient: positive

End Behavior:

End Behavior of a Polynomial Function

Degree: even

Leading Coefficient: negative

End Behavior:

Degree: odd

Leading Coefficient: negative

End Behavior:

It is an even-degree polynomial function.

The graph intersects the x-axis at two points, so the function has two real zeros.

It is an odd-degree polynomial function.

The graph has one real zero.

The term a is assumed to be non-zero and is called the leading coefficient.

n