## The Factor Theorem

The Factor Theorem

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

Example: Show thatis a factor of

henceis a factor of

Example:andare both roots of the quadratic expressionFind

Sinceandare both rootsandare both factors. A quadratic equation has at most two real factors/roots hence

More complicated questions may involve simultaneous equations:

andare both roots ofFind a and b and factorise f(x)..

divided byremainder is

divided byremainder is

We now solve the simultaneous equations

(1)

(2)

3*(1)+(2) gives

Then from (1)

is a cubic and has the two given roots so must have a third linear factor

by considering the coefficient ofand by considering the constant term 1