The Remainder and Factor Theorems

In the first factoring section, we discussed several techniques for factoring polynomials that, in terms of their degree and number of terms, were relatively small. Unfortunately, the situation for larger polynomials isn't as a clear cut. The Abel-Ruffini Theorem says that there are no formulas like the Quadratic Formula for polynomials whose degree is greater than four. There are, however, some theorems and techniques we can use to find a list of potential factors and then test to see which ones actually are.

The Remainder Theorem

If $p(x)$ is a polynomial function then the remainder when $p(x)$ is divided by $x - c$ is $p(x)$.

The Factor Theorem

If $p(x)$ is a polynomial function then the remainder when $x - c$ is a factor of p if and only if $p(c) = 0$.

Video Lectures

Lectures

The Polynomial Remainder Theorem, Derived

The Remainder Theorem gives us a way to determine what the remainder will be when a polynomial is divided by a linear factor without going to the trouble of doing the long division. That's good to know but it's really helpful in proving the Factor Theorem which gives us a relationship between the factors of a polynomial and its roots. In this lecture, we're going to both state and prove both of those theorems. (lecture slides)

The Factor and Remainder Theorems, Derived

Individual Examples

The Remainder Theorem - Examples

In this lecture, we're going to look at some examples that illustrate how the Remainder Theorem can be used to answer questions about polynomials.

The Factor Theorem - Examples

In these examples, we're going to see how to use the Factor Theorem to answer a key question when working with polynomials an polynomial equations: How can you tell whether or not one polynomial is a factor of another.