Understanding Descartes' Rule of Signs

Descartes' Rule of Signs

Ever wondered how to quickly determine how many solutions a polynomial equation might have? Descartes' Rule of Signs does exactly that by examining the pattern of coefficient signs.

For positive real zeros, count the number of sign changes in the polynomial's coefficients. The number of positive real roots equals this count or less by an even number. For example, in f$x$ = x^5 + 4x^4 - 3x² + x - 6, we see 3 sign changes (+ to -, - to +, + to -), so there are either 3 or 1 positive real roots.

For negative real zeros, substitute -x for x in your polynomial to create f$-x$, and then count the sign changes. In our example, f$-x$ = -x^5 + 4x^4 - 3x² - x - 6, which becomes -x^5 + 4x^4 - 3x² - x - 6. This has 2 sign changes, meaning there are either 2 or 0 negative real roots.

Quick Tip: To find all possibilities for the number of roots, start with the number of sign changes and count down by 2 until you reach 1 or 0. For our example, the positive roots could be 3 or 1, while negative roots could be 2 or 0.