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Chapter 5 : Polynomial Functions

Here are a set of practice problems for the Polynomial Functions chapter of the Algebra notes.

  1. If you’d like a pdf document containing the solutions the download tab above contains links to pdf’s containing the solutions for the full book, chapter and section. At this time, I do not offer pdf’s for solutions to individual problems.
  2. If you’d like to view the solutions on the web go to the problem set web page, click the solution link for any problem and it will take you to the solution to that problem.

Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section.

Here is a list of all the sections for which practice problems have been written as well as a brief description of the material covered in the notes for that particular section.

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.