# Additional Factoring Methods

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

## The Factor Theorem

*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,*a*_{0}*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.