The Factor and Remainder Theorems

Long division with polynomials sounds, and is, a great deal more complicated than long division with numbers. Fortunately though, it is not always necessary. There are two very helpful theorems which often turn the problem of long division into one of substitution.

The Factor Theorem:

Ifis a factor ofthensois also a root ofor equivalently, a solution of the equation

The Factor Theorem is a special case of The Remainder Theorem.

The Remainder Theorem

The remainder when performing the long division ofbyis .Ifis a factor ofthen

Example: Show thatis a factor of

henceis a factor of

Example. Find the remainder whenis divided by

We calculateNoteis the solution to

More complicated questions may involve simultaneous equations:

Whenis divided bythe remainder is 4. Whenis divided bythe remainder is 6. Find a and b.

divided byremainder is 4

divided byremainder is 6

We now solve the simultaneous equations

(1)

(2)

3*(1)+(2) gives

Then from (1)