Remainder Theorem

How to use the remainder theorem to find the remainder of the division of a polynomial: theorem, 2 examples, and their solutions.

Theorem

Theorem

The remainder of f(x)/(x - a) is
f(a).

This is the remainder theorem.

Example 1

Example

Solution

Set f(x) = x3 - 7x + 11.
Then (given) = f(x)/(x - 2).

The zero of (x - 2) is 2.

Then, by the remainder theorem,
the remainder of f(x)/(x - 2) is
f(2).

f(x) = x3 - 7x + 11

Then
f(2) = 23 - 7⋅2 + 11.

23 = 8
-7⋅2 = -14

8 - 14 = -6

-6 + 11 = 5

So the remainder is 5.

Example 2

Example

Solution

Set f(x) = 2x4 + x3 - 5x2 + 3x + 4.
Then (given) = f(x) ÷ (x + 1).

The zero of (x + 1) is -1.

Then, by the remainder theorem,
the remainder of f(x) ÷ (x + 1) is
f(-1).

f(x) = 2x4 + x3 - 5x2 + 3x + 4

Then
f(-1) = 2⋅(-1)4 + (-1)3 - 5⋅(-1)2 + 3⋅(-1) + 4.

(-1)4 = 1
+(-1)3 = +(-1) = -1
(-1)2 = 1
+3⋅(-1) = -3

2⋅1 = 2
-5⋅1 = -5

-1 - 3 + 4 = 0

2 - 5 = -3

So the remainder is -3.