Descartes rule of signs helps in finding the number of roots or zeros of a polynomial. This rule is a technique for determining the number of positive real roots and negative real roots of a polynomial. It gives the upper bound of positive and negative roots but it does not give the exact number of roots. The number of positive real roots of a polynomial is the number of changes of sign in its coefficients. The number of negative real roots of a polynomial is the number of changes of sign in its coefficients of f (-x).
Problem 1: How many numbers of positive roots in the polynomial: -5x^5 + 3x + 2x^2 – 2
Solution: Given polynomial: -5x^5 + 3x + 2x^2 – 2
=> Rewrite the polynomial from highest to lowest exponent: -5x^5 + 2x^2 + 3x – 2
=> Now find the number of changes in sign. That is from minus to plus sign or plus to minus sign.
=> From the given polynomial there are 2 changes -5x^5 + 2x^2 and + 3x – 2
=> So, there will be at most 2 positive roots.
Problem 2: Determine the number of negative roots f(x) = x^5 + x^4 + 2x^3 + 5x^2 + x + 3
Solution: To find negative roots find f (-x)
=> f (-x) = (-x)^5 + (-x)^4 + 2(-x)^3 + 5(-x)^2 +(- x) + 3
= -x^5 + x^4 - 2x^3 + 5x^2 - x + 3
=> There are five sign changes. So, there will be at most 5 negative roots.