Remainder Theorem

Remainder Theorem

How to solve the remainder theorem problems: theorem, examples, and their solutions.

Theorem

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

Let's think of f(x) ÷ (x - a).

Then f(x) = (x - a)⋅(quotient) + (remainder).

If x = a,
f(a) = (a - a)⋅(quotient) + (remainder)
= (remainder).

So f(a) is the remainder.

So f(x) = (x - a)(quotient) + f(a).

And the remainder of f(x) ÷ (x - a) is f(a).

This is the remainder theorem.

Example 1

Find the remainder of the given expression. (x^3 - 7x + 11)/(x - 2)

The zero of the divisor (x - 2) is 2.

So the remainder is f(2).

Example 2

Find the remainder of the given expression. (2x^4 + x^3 - 5x^2 + 3x + 4)/(x + 1)

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

So the remainder is f(-1).

Example 3

If the remainder of the given expression is 11, find the value of a. (x^5 - 8x^4 + 7x^2 - 3x + a)/(x - 1)

The zero of the divisor (x - 1) is 1.

So the remainder is f(1).

f(1) = -3 + a

And it says the remainder is 11.

So f(1) = -3 + a = 11.