Algebra - Polynomial Functions

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Chapter 5 : Polynomial Functions

In this chapter we are going to take a more in depth look at polynomials. We’ve already solved and graphed second degree polynomials (i.e. quadratic equations/functions) and we now want to extend things out to more general polynomials. We will take a look at finding solutions to higher degree polynomials and how to get a rough sketch for a higher degree polynomial.

We will also be looking at Partial Fractions in this chapter. It doesn’t really have anything to do with graphing polynomials but needed to be put somewhere and this chapter seemed like as good a place as any.

Here is a brief listing of the material in this chapter.

Dividing Polynomials – In this section we’ll review some of the basics of dividing polynomials. We will define the remainder and divisor used in the division process and introduce the idea of synthetic division. We will also give the Division Algorithm.

Zeroes/Roots of Polynomials – In this section we’ll define the zero or root of a polynomial and whether or not it is a simple root or has multiplicity \(k\). We will also give the Fundamental Theorem of Algebra and The Factor Theorem as well as a couple of other useful Facts.

Graphing Polynomials – In this section we will give a process that will allow us to get a rough sketch of the graph of some polynomials. We discuss how to determine the behavior of the graph at \(x\)-intercepts and the leading coefficient test to determine the behavior of the graph as we allow x to increase and decrease without bound.

Finding Zeroes of Polynomials – As we saw in the previous section in order to sketch the graph of a polynomial we need to know what it’s zeroes are. However, if we are not able to factor the polynomial we are unable to do that process. So, in this section we’ll look at a process using the Rational Root Theorem that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes.

Partial Fractions – In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression. What we will be asking here is what “smaller” rational expressions did we add and/or subtract to get the given rational expression. This is a process that has a lot of uses in some later math classes. It can show up in Calculus and Differential Equations for example.