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Descartes' Rule Of Signs

Descartes' Rule Of Signs:

Given a polynomial equation with real coefficients:

1. The number of positive real solutions to the given equation is either equal to the number of variations in the sign of the polynomial or two less than the number of the variations in the sign of the polynomial.

From the same polynomial:

with x replaced with -x; the number of negative real solutions to the polynomial equation is either equal to the number of variations in the sign of the polynomial or two less than the number of the variations in the sign of the polynomial.

number of variations is the number of sign changes from one term to the other.

ex.
x3- 3x2+ 4x - 5 = 0

There are three different sign changes in the equation.

(+ x3 to - 3x2 is a sign change or variation)

therefore there are either three or one real positive solution to the equation.

Replace x with -x in the equation gives:

(-x)3 - 3(-x)2+ 4(-x) - 5 = 0
-x3- 3x2- 4x - 5 = 0

There are no variations in the sign, therefore there are no real negative solutions to the equation.

Subject: 
Algebra
Subject X2: 
Algebra

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