Descartes' Rule of Signs

Theorem: A polynomial has no more positive roots than it has sign changes.

The idea of a sign change is a simple one. Consider the polynomial

P(x) = x ³ – 8 x ² + 17 x – 10

Proceeding from left to right, we see that the terms of the polynomial carry the signs  + – + –  for a total of three sign changes. Descartes' Rule of Signs tells us that this polynomial may have up to three positive roots. In fact, it has exactly three positive roots: At  1, 2, and  5 .

Just as the Fundamental Theorem of Algebra gives us an upper bound on the total number of roots of a polynomial, Descartes' Rule of Signs gives us an upper bound on the total number of positive ones. A polynomial may not achieve the maximum allowable number of roots given by the Fundamental Theorem, and likewise it may not achieve the maximum allowable number of positive roots given by the Rule of Signs.

For example, the polynomial  f(x) = x ³ – 3 x ² + 3 x – 1  has three sign changes but just one positive root (of multiplicity 3) at  x = 1 . The polynomial  g(x) = x ³ – x ² + 1  has two sign changes but no positive roots.

A corollary to the Rule of Signs says

A polynomial  P  has no more negative roots than  P(–x)  has sign changes.

To use this, we just replace every instance of  x  with  –x  in our polynomial. This essentially changes the signs of the odd powers but not the even ones. Thus:

P(x) = x ³ – 8 x ² + 17 x – 10

P(–x) = –x ³ – 8 x ² – 17 x – 10

P(–x)  has no sign changes and so  P(x)  has no negative roots (as observed above).

On the other hand,

g(x) = x ³ – x ² + 1

g(–x) = –x ³ – x ² + 1

g(–x)  has one sign change and, indeed, g(x)  has one negative root (and, as observed above, no positive ones). The Rational Root Theorem shows us that this root must be an irrational number.