Descartes' Rule of Signs

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Descartes' Rule of Signs

Given a polynomial such as:
x⁴ + 7x³ - 4x² - x - 7
it is possible to say anything about how many positive real roots it has, just by looking at it?

Here's a striking theorem due to Descartes in 1637, often known as "Descartes' rule of signs": The number of positive real roots of a polynomial is bounded by the number of changes of sign in its coefficients. Gauss later showed that the number of positive real roots, counted with multiplicity, is of the same parity as the number of changes of sign.

Thus for the polynomial above, there is at most one positive root, and therefore exactly one.

In fact, an easy corollary of Descartes' rule is that the number of negative real roots of a polynomial f(x) is determined by the number of changes of sign in the coefficients of f(-x). So in the example above, the number of negative real roots must be either 1 or 3.

The Math Behind the Fact:
A proof of Descartes' Rule for polynomials of arbitrary degree can be carried out by induction. The base case for degree 1 polynomials is easy to verify! So assume the p(x) is a polynomial with positive leading coefficient. The final coefficient of p(x) is given by p(0).

If p(0)>0, then the number of sign changes must be even, since the first and last coefficient of p(x) are both positive. Moreover, the number of roots (counted with multiplicity) must also be even, since p(x) is also positive for very large x, so the graph of p(x) can only cross the x-axis an even number of times. Similar arguments show that if p(0)<0, then the number of sign changes is odd and the number of positive roots is odd. Thus the number of sign changes and number of roots have the same parity.

If p(x) had more roots than sign changes then it must have at least 2 more roots. But p'(x) is a polynomial with zeroes between each of the roots of p(x) [why?], so p'(x) has at least 1 more root than sign changes of p(x). This yields a contradiction because p'(x) has no more sign changes than p(x) does, and the inductive hypothesis then implies that p'(x) has no more roots than sign changes of p(x).

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Descartes' Rule of Signs