The Remainder Theorem is the generalized form of The Factor Theorem.

The Remainder Theorem

When a polynomial expressionis divided by a linear factorthe remainder isIfthenis a factor of

Example:

Show thatis a factor of

thereforeis a factor.

Example

If the remainder whenis divided byis 3 and the remainder whenis divided byis 4, findand

Remainder ofon division byis

Remainder when p(x) is divided byis

We solve the simultaneous equations

(1)

(2)

(1)+2*(2) givesthen from (2)

Then