Descartes Rule of Signs

René Descartes was a French mathematician and a philosopher. He is mostly known by its coordinate system and for setting the grounds to the modern geometry. He also studied polynomials and in 1637 gave an important theorem known as Descartes’ Rule of Signs.

Descartes Rule of Signs says that the number of positive real roots of a polynomial is bounded by the number of changes of sign in its coefficients.

Lemma: The number of negative real roots of $f(x)$ is the same as the number of changes in the coefficients of the terms of $f(x)$.

Example 10. Determine the number of positive and negative real roots for the given function:

$f(x) = x^5 + 5x^4 – 6x^2 + 2x – 16$

First thing you need to pay attention to is whether this equation is arranged in descending powers of the variable. If it’s not, you should change the order of the variables so that it is.

Now take a look at the coefficients.

1               5             -6              2               -16

We have three changes of signs. From positive to negative between 5 and -6, from negative to positive between -6 and 2 and from positive to negative between 2 and -16. This means that we can have 3 real positive zeros or less but an odd number of zeros. This means that the number of positive zeros is either 3 or 1.

Using the Lemma, we can find possible number of negative real roots.

$f(-x) = -x^5 + 5x^4 – 6x^2 – 2x – 16 = 0$

Here we have two sign changes, which means that either we have two negative zeros or less but even number of zeros, which means that there can be 2 or 0 negative zeros.