Descartes Rule of Signs

Definition: Descartes’ rule of signs can be used to determine how many positive and negative real roots a polynomial has. It involves counting the number of sign changes in f(x) for positive roots and f(-x) for negative roots. The number of real roots may also be given by the number of sign changes minus an even integer.

Example: Use the rule of signs to determine the number of positive and negative real roots:

f(x) = x³ + 2x² + 5x + 4

Solution:

Step 1: To find the number of possible positive real zeros, count the number of sign changes in f(x). Since all the terms are positive, we can conclude that there are no positive real zeros.

Step 2: To find the number of possible negative real zeros, count the number of sign changes in f(-x). To get f(-x), we can replace x with (-x) in the given polynomial:

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

Since the signs in the function alternate between (-) and (+) three times, we know the number of negative real zeros of this function is either 3, or 3 minus an even integer.
This tells us there are either three real negative roots, or (3 – 2 = 1) one negative real root.

The polynomial has no positive real roots and either 3 or 1 negative real roots.

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