Additional Factoring Methods

Quick Navigation Video Lectures

Solving polynomial equations ultimately comes down to finding a way to factor them. Then you just apply the Zero Product Rule, solve some smaller equations and you get your final results. With relatively small polynomials, e.g. quadratics, there are methods you can use to always get the final factorization. In this section, we're going to look at some tools you can use to try to find a factorization when those more straightforward methods don't work. The methods are summarized below then you'll see specific examples of how they're used in the video lectures.

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 $p(c) = 0$.

Descartes's Rule of Signs

Let $p(x)$ be a polynomial written in the standard form. The number of positive real roots of p is equal to either the number of sign changes in the coefficients or that number minus an even integer.

The number of negative roots is equal to the number of sign changes in the coefficients of $p(-x)$ or that number minus an even integer.

The Rational Root Theorem

Suppose $p(x)$ is a polynomial with integer coefficients. Then every rational root $\frac{p}{q}$, written in lowest terms, meets two conditions:

p is a factor of the constant term, a₀
q is a factor of the leading coefficient, a_n

The Complex Conjugate Root Theorem

If z is a root of a polynomial with real coefficients then $\bar{z}$ is also a root.

Video Lectures

Descartes's Rule of Signs

In this lecture, we'll see how to use Descartes's Rule of Signs to narrow down the possible roots of a polynomial. (lecture slides)

The Rational Root Theorem

In this lecture, we're going to look at some examples of how the Rational Root Theorem can be used to get a list of all the possible rational roots of a polynomial. (lecture slides)

Complex Conjugate Root Theorem

The Complex Conjugate Root Theorem says that if z is a complex root of a polynomial then the conjugate of z is also a root. This video walks through the proof by first showing four smaller statements that demonstrate some basic methods for proving statements about complex variables.

Finding Roots, Part 1

Now that we've added several new tools to our factoring toolbox, I want to look at some specific examples so you can see how they all work together.

Finding Roots, Part 2

In this final lecture in our discussion of factoring polynomials, we're going to look at two more examples that illustrate two interesting situations.