10.1 - Introduction to Polynomials

Polynomials are an extension of quadratics so you may it useful to review quadratics before reading this section. Also many of the ideas discussed here involve complex numbers so you may want to review those as well.

To create a polynomial imagine carrying out the following steps: The result is a polynomial. Note that some of the coefficients could be zero so that some of the powers of x could be absent. Here is the formal definition of a polynomial:


Definition: A polynomial is an expression of the form:
an · x n + an −1 · x n −1 +  …  + a2 · x 2 + a1 · x + a0,
where x is a variable, n is a positive integer and a0, a1, … , an −1, an are constants. The highest power of x that occurs is called the degree of the polynomial. The terms are usually written in order from highest power of x to lowest power.




Examples:

Graph of a polynomial

We can create a polynomial function, called say f, whose input is x and whose output, f (x), is the polynomial evaluated at x:
an x n + an −1 x n −1 +  …  + a2 x 2 + a1 x + a0

We can then call the output of the function “y” and make a graph of y versus x. We will get a curve like the two curves shown to the right. The graph of a polynomial function oscillates smoothly up and down several times before finally “taking off for good” in either the up or down direction.

The degree of the polynomial gives the maximum number of “ups and downs” that the graph of the polynomial can have. It also gives the maximum number of crossings of the x axis that the polynomial can have.


Polynomial equation

If we set the polynomial equal to zero or if we set y = 0 or f (x) = 0 then we get a so-called polynomial equation:
an x n + an −1 x n −1 +  …  + a2 x 2 + a1 x + a0 = 0.
(Note that setting y = 0 in the polynomial’s graph means that we are looking at points where the graph crosses the x axis, and setting f (x) = 0 in the polynomial function means that we are looking for values of x for which the output of the polynomial function is zero.



There is a close connection between: This connection is made formal by the Factor theorem and the Fundamental theorem of algebra.


The factor theorem

Let f (x) be a polynomial.
  • If (xr) is a factor of the polynomial, then r is a root of the polynomial equation f (x) = 0.
  • Conversely, if the polynomial equation f (x) = 0 has a root r, then (xr) is a factor of the polynomial f (x).



The fundamental theorem of algebra

Over the complex numbers, a polynomial equation of degree n has exactly n roots. Over the real numbers it may have less than n.
  • Some of the roots may be real.
  • Some of the roots may be complex. If so then their complex conjugates will also be roots.
  • Some of the roots may be equal (these are called multiple or repeated roots).


Notes on the Factor theorem and the Fundamental theorem of algebra:


Example: This example is meant to illustrate the various quantities related to polynomials that were defined above as well as these two theorems.


Example: Consider the polynomial equation x 4 − 8 x 2 = 0. The left-hand-side can be factored as x 2 (x + 2.83) (x − 2.83). If we write the equation like this:
(x − 0) 2 (x + 2.83) (x − 2.83) = 0,
then we see that it has roots at x = 2.83 and x = −2.83, as well as a double root at x = 0. This means that the graph of the polynomial function y = x 4 − 8 x 2 should cross the x axis at x = −2.83 and x = 2.83 and it should touch the x axis at x = 0. This can be verified by looking at the red curve shown above.