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.
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.
- 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.
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).
- 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
x > 0 and is a decreasing function for x < 0.
- Using a graphing calculator to find the roots of polynomials: