Question

use the remainder theorem to find the remainder when is divided by x-3 then use the factor theorem to determine whether is a factor of

Answer 1

To find the remainder when a polynomial is divided by x-3, evaluate the polynomial at x=3. If the result is 0, x-3 is a factor of the polynomial according to the Factor Theorem.

Step-by-step explanation:

Understanding the Remainder and Factor Theorems

The Remainder Theorem states that if a polynomial f(x) is divided by x - a, the remainder is f(a). To use this theorem, you simply evaluate the polynomial at x = a.

For instance, if we're given a polynomial f(x) and we want to find the remainder when it is divided by x - 3, we evaluate f(3). This value is the remainder.

The Factor Theorem is closely related to the Remainder Theorem and tells us that if f(a) = 0, then x - a is a factor of the polynomial f(x). After finding the remainder using the Remainder Theorem, if the remainder is 0, then x - 3 is a factor of f(x).

If the student provides a specific polynomial, apply these steps to find the remainder and determine factorship:

1. Evaluate the polynomial at x = 3.
2. If the result is 0, conclude that x - 3 is a factor of f(x).
3. If the result is a non-zero number, that number is the remainder when f(x) is divided by x - 3.