Lesson Objectives
• Learn how to evaluate a polynomial function using the remainder theorem

## How to Evaluate a Polynomial Function Using the Remainder Theorem

In this lesson, we will learn about the remainder theorem. The remainder theorem gives us a shortcut for evaluating a polynomial function at a given value.
We can break up any polynomial function using the following formula: $$f(x)=(x - k)q(x) + r$$ (x - k) is the divisor
q(x) is the quotient
r is the remainder
When our polynomial function is written in this format, it is obvious that f(k) = r: $$f(x)=(x - k)q(x) + r$$ $$f(k)=(k - k)q(x) + r$$ $$f(k)=0 + r=r$$ This result is known as the remainder theorem. This tells us that if the polynomial function f(x) is divided by (x - k), the remainder is the same as f(k). We can use this to quickly evaluate a polynomial function for a given value. Let's look at an example.
Example #1: Evaluate f(x) at k. $$f(x)=11x^3 + 6x^2 - 14$$ $$k=-1$$ First, let's plug in a -1 for x and see what we get: $$f(-1)=11(-1)^3 + 6(-1)^2 - 14$$ $$f(-1)=-11 + 6 - 14=-19$$ Now, we will see that we get the same result as the remainder from synthetic division. Notice how the remainder is -19. This tells us that f(-1) = -19.

#### Skills Check:

Example #1

Evaluate f(x) at k. $$f(x)=13x^3 + 7x^2 - x - 6$$ $$k=1$$

A
$$f(k)=-5$$
B
$$f(k)=9$$
C
$$f(k)=10$$
D
$$f(k)=15$$
E
$$f(k)=13$$

Example #2

Evaluate f(x) at k. $$f(x)=x^3 + 19x^2 + 74x + 56$$ $$k=-2$$

A
$$f(k)=8$$
B
$$f(k)=10$$
C
$$f(k)=-11$$
D
$$f(k)=-24$$
E
$$f(k)=19$$