Rational Zero Theorem

Definition

The Rational Zero Theorem provides a method to identify possible rational zeros of a polynomial function. It states that any rational zero of the polynomial function with integer coefficients is a ratio of factors of the constant term to factors of the leading coefficient.

5 Must Know Facts For Your Next Test

1. A rational zero is expressed in the form $\frac{p}{q}$ where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.
2. The theorem only applies to polynomials with integer coefficients.
3. The Rational Zero Theorem generates a list of potential rational zeros, which must be tested to determine if they are actual zeros.
4. If $P(x)$ has integer coefficients and $a_0$ as its constant term, then every rational solution $\frac{p}{q}$ satisfies that $p$ divides $a_0$ and $q$ divides $a_n$, where $a_n$ is the leading coefficient.
5. The Rational Zero Theorem can help narrow down the possible candidates for synthetic division or other root-finding techniques.

Review Questions

• What does the Rational Zero Theorem state about possible rational zeros?
• How do you determine the potential rational zeros for a given polynomial?
• Can the Rational Zero Theorem be applied to polynomials with non-integer coefficients?

Related terms

Polynomial Function: A mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.

Synthetic Division: A simplified form of polynomial division, particularly useful for dividing by linear factors.

Factor Theorem:

$x - r$ is a factor of polynomial $P(x)$ if and only if $P(r) = 0$. This theorem is often used in conjunction with the Rational Zero Theorem.