Remainder Theorem

In this lesson, we will look into the Remainder Theorem.

Recall that for long division for integers, the dividing process stops when the remainder is less than the divisor.

dividend = divisor × quotient + remainder

The process is similar for division of polynomials. The dividing stops when the remainder is less that the degree of the divisor.

The Remainder Theorem states that

If a polynomial f(x) is divided by a linear divisor x – a, the remainder is f(a)

Hence, when the divisor is linear, the remainder can be found by using the Remainder Theorem.

Example:

Find the remainder when 4x³ – 5x + 1 is divided by

a) x – 2
b) x + 3
c) 2x – 1

Solution:

Let f(x) = 4x³– 5x + 1

a) When f(x) is divided by x – 2, remainder,

R = f(2) = 4(2)³– 5(2) + 1 = 23

b) When f(x) is divided by x + 3, remainder,

R = f(–3) = 4(–3)³– 5(–3) + 1 = –92

c) When f(x) is divided by 2x – 1, remainder,

Example:

The expression 4x² – px + 7 leaves a remainder of –2 when divided by x – 3. Find the value of p.

Solution:

Let f(x) = 4x²– px + 7

By the Remainder Theorem,

f(3) = –2

4(3)²– 3p + 7 = –2

p = 15

Videos

The remainder theorem -
Professor Edward Burger explains the remainder theorem

More on the remainder theorem -
Professor Edward Burger explains more on the remainder theorem

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