Polynomial Functions of Higher Degree

Polynomial Functions of Higher Degree

The degree of a polynomial is the highest exponent in the function.

The degree of function f is 5. This is because the term with the greatest exponent is raised to the power of 5.

The graph of a polynomial function is continuous. This means that the graph of a polynomial function has no breaks, holes, or gaps. 

The polynomial functions that have the simplest graphs are monomials of the form 

where n is an integer greater than zero.

Even and Odd Polynomial Functions

Even

If n is even, the graph of touches the axis at the x-intercept.

Odd

                                                                                     

If n is odd, the graph of crosses the axis at the x-intercept.

The greater the value of  n, the flatter the graph is on the interval (-1,1).

When n is odd: the ends of the graph are in opposite directions.

If the leading coefficient is positive, the graph falls to the left and rises to the right.

If the leading coefficient is negative, the graph rises to the left and falls to the right.

Even

When n is even: the ends of the graph are each in the same direction.

If the leading coefficient is positive, the graph rises to the left and right.

If the leading coefficient is negative, the graph falls to the left and right.

Zeroes of Polynomial Functions

It can be shown that for a polynomial function f of degree  n, the following statements are true:
The graph of f has at most n real zeroes.
The function f has at most n-1 relative extrema.

An extremum is a relative maximum or minimum on an interval.

The three extrema for a polynomial of degree 4.

The two extrema for a polynomial of degree 3.

A single extremum for a polynomial of degree 1.

Works Cited

"Experimental Feature." Wolfram|Alpha: Computational Knowledge Engine. N.p., n.d. Web. 12 Jan. 2015.

"Extremum." -- from Wolfram MathWorld. N.p., n.d. Web. 13 Jan. 2015.

Larson, Ron, Robert P. Hostetler, Bruce H. Edwards, and David E. Heyd. Precalculus with Limits: A Graphing Approach. Boston: Houghton Mifflin, 1997. Print.

X. "Sec 3.2 Polynomial Functions and Their Graphs." (n.d.): n. pag. Section 3.2 Polynomial Functions and Their Graphs. New York University. Web.
Stapel, Elizabeth. "Degrees, Turnings, and "Bumps"" Degrees, Turnings, and "Bumps" N.p., n.d. Web. 13 Jan. 2015.