Factor Theorem

Definition

The Factor Theorem states that a polynomial $f(x)$ has a factor $(x - c)$ if and only if $f(c) = 0$. It is used to determine whether a given binomial is a factor of the polynomial.

5 Must Know Facts For Your Next Test

1. If $f(c) = 0$, then $(x - c)$ is a factor of the polynomial $f(x)$.
2. The Factor Theorem is a special case of the Remainder Theorem.
3. It provides a method for finding zeros (roots) of polynomials.
4. To apply the Factor Theorem, substitute the potential zero into the polynomial and check if it equals zero.
5. The theorem can be extended to find all factors of higher-degree polynomials by repeated application.

Review Questions

• What does it mean if $f(c) = 0$ for a polynomial $f(x)$?
• How do you use the Factor Theorem to determine if $(x - c)$ is a factor of $f(x)$?
• Explain how the Factor Theorem helps in finding all zeros of a polynomial.

Related terms

Remainder Theorem: If a polynomial $f(x)$ is divided by $(x - c)$, the remainder is $f(c)$. This theorem generalizes the Factor Theorem.

Polynomial Long Division: A method used to divide one polynomial by another, yielding both quotient and remainder.

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