Descartes's Rule of Signs
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.
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.
