Descartes' Rule of Signs

Let f(x) define a polynomial function with real coefficients and a nonzero constant term, with terms in descending powers of x.

The number of positive real zeros of f either equals the number of variations in sign occurring in the coefficients of f(x), or is less than the number of variations by a positive even integer.
The number of negative real zeros of f either equals the number of variations in sign occurring in the coefficients of f(−x), or is less than the number of variations by a positive even integer.

Example

Determine the different possibilities for the number of possible positive, negative, and nonreal complex zeros of $LaTeX: f\left(x\right)=x^4-6x^3+8x^2+2x-1$ $f\left(x\right)=x^4-6x^3+8x^2+2x-1$ .

Solution

We first consider the possible number of positive zeros by observing that f(x) has three variations in signs.

Variations in Signs of f(x).PNG

Thus, by Descartes' Rule of Signs, f(x) has either 3 or 3-2 = 1 positive real zeros.

For negative zeros, consider the variations in signs of f(-x):

Since there is one variation in sign, f(x) has exactly one negative real zero.

One possible combination of the zeros is one negative real zero, three positive real zeros, and no nonreal complex zeros.
Another possible combination of the zeros is one negative real zero, one positive real zero, and two nonreal complex zeros.

Example

This video introduces Descartes' Rule of Signs and goes through an example using it.

Please complete Homework 3.3 on the next page.