Descartes’ Rule of Signs

Definition

Descartes’ Rule of Signs provides a method to determine the number of positive and negative real zeros in a polynomial function. It examines the number of sign changes in the sequence of coefficients.

5 Must Know Facts For Your Next Test

1. The number of positive real zeros is either equal to the number of sign changes in $f(x)$ or less than that by an even number.
2. The number of negative real zeros is determined by applying Descartes' Rule to $f(-x)$.
3. A polynomial with no sign changes in $f(x)$ has no positive real zeros.
4. If $f(-x)$ has no sign changes, there are no negative real zeros.
5. Descartes' Rule does not provide information about complex roots or multiple roots.

Review Questions

• How many positive real zeros can a polynomial have if it has three sign changes?
• What steps must you follow to determine the possible number of negative real zeros using Descartes’ Rule?
• Can Descartes’ Rule of Signs indicate the exact number of real zeros?

Related terms

Polynomial Function: A function that is represented by a polynomial equation involving variables and coefficients.

Real Zeros: The values of x for which a polynomial equals zero and are real numbers.

$f(-x)$: $f(-x)$ represents the polynomial function evaluated at $-x$ and is used to determine the number of negative real zeros.