Objectives: In this tutorial, we define polynomial functions. We investigate some properties of polynomials including the domain, range, roots and symmetry. The graphs of polynomials of degree n and the polynomial xn are also investigated. After working through these materials, the student should be able

• to recognize a polynomial function;
• to describe the "long-term behavior" of polynomial functions;
• to distinguish between xn and xn when n and m are different;
• to recall the Fundamental Theorem of Algebra and other results on the roots of polynomial functions;
• to use a calculator to find the roots of a polynomial function.

Modules:

 Definition. A polynomial function f is a function of the form f(x) = anxn + an-1xn-1 + ... + a2x2 + a1x1 + a0 where a0, a1, ..., an are real numbers. If an is not zero, then f is said to have degree n.

• Discussion [Using Flash]

• Examples.

• If you zoom out sufficiently far, then the graph of the polynomial function f(x) = anxn + an-1xn-1 + ... + a2x2 + a1x1 + a0 looks just like the polynomial function g(x) = anxn. Use this LiveMath notebook to experiment to see that this is true.

• You should become familiar with the differences of the graphs of the polynomials f(x) = xn and g(x) = xm for n not equal to m. Use this LiveMath notebook to experiment with different values of n and m.

 Definition. A function f is increasing on the set B if for all c and d in B, c < d implies f(c) < f(d). A function f is decreasing on the set B if for all c and d in B, c < d implies f(c) > f(d).

• Examples:

• If n is odd then f(x) = xn is an increasing function.
• If n is even then f(x) = xn is an increasing function for > 0 and is a decreasing function for < 0.

• Using a graphing calculator to find the roots of polynomials: