Descartes' rule of signs

The sign rule of Descartes in mathematics - similar to the Sturm sequence - used to determine the maximum number of positive and negative zeros of a real polynomial.

Rule

The sign rule of Descartes is:

An important conclusion follows:

It is named after the French philosopher and mathematician René Descartes, who has described it in 1637 in his work La Géométrie first.

Examples

Maximum number of positive zeros

In the polynomial

Changes the sign of the coefficient three times. According to Descartes thus has the polynomial f (x ) 3 positive zeros or 3 - 2 = 1 positive root. In fact, the polynomial f (x) has a positive zero.

Maximum number of negative zeros

To determine the maximum number of the negative zero points, first from the polynomial f (x) is a new polynomial f ( x ) is formed. This means that the signs of the coefficients for odd exponent be changed while the signs of the coefficients for an even exponent remain unchanged. The sign rule of Descartes is then applied to this new polynomial.

Referring again to the polynomial

That is the new polynomial

Here the sign of the coefficient changes four times. According to Descartes thus has the polynomial f (x ) is either 4, 4-2 = 2 or 4-4 = 0 negative zeros. In fact, the polynomial f (x) has no negative zero.