Question

Which theorem states that if the value of a polynomial is zero when x = k, then x-k is a factor of the original polynomial?

A. Remainder theorem
B. Factor theorem
C. Rational root theorem
D. Intermediate value theorem

Answer 1

The Factor theorem states that if a polynomial evaluates to zero when x = k, then x-k is a factor of the original polynomial.

Step-by-step explanation:

The correct theorem that states if the value of a polynomial is zero when x = k, then x-k is a factor of the original polynomial is the Factor theorem.

This theorem is based on the fact that if a polynomial P(x) evaluates to zero when x = k, then P(k) = 0. This means that (x - k) is a factor of P(x).

For example, let's say we have a polynomial P(x) = x^3 - 4x^2 + 5x - 2, and we want to find its factors. If P(k) = 0 when k = 2, then (x - 2) is a factor of P(x). We can use polynomial division or synthetic division to verify this.