To determine the number of positive and negative real roots, we use a technique founded by Rene Descartes, thus namedDecartes' Rule of Signs.
To find the number of positive real roots, start with the sign of the coefficient of the term with the highest power of the
variable. Count the sign changes as you proceed through the polynomial. The number of sign changes is the number of
positive real roots or less than it by a multiple of two. For example, if there are 2 sign changes there are either 2 or (2 – 2) =
0 positive real roots.
Example:Find the number of positive real roots of f(x) = -4x^5 – 11x^4 + 2x^3 + 9x^2 - x + 3.
Starting with -4x^5 , there is a sign change at 2x^3, another at -x and a third at 3. Therefore there are 3 sign changes and either 3 or (3 – 2) = 1 positive real roots.
To find the number of negative real roots of f(x), substitute -x into f(x) to get f(-x). Then proceed in the same manner as you would when finding the number of positive real roots. In the above example,
f(-x) = -4(-x)^5 – 11(-x)^4 + 2(-x)^3 + 9(-x)^2 - (-x) + 3
f(-x) = 4x^5 - 11x^4 – 2x^3 + 9x^2 + x + 3
Starting with 4x^5 , there is a sign change at 11x^4 and another at 9x^2 . Therefore, there are 2 or (2 – 2) = 0 negative real roots.
• Note that the number of roots in a polynomial function equals the highest power. In the above example, there are 5
roots. The sum of the positive and negative roots will not always equal the highest power. In such cases, thosepolynomial functions have imaginary roots