We can use the factor theorem to find if a particular value ofis a root of a polynomial equation or to find out if a particular linear factor divides a polynomial perfectly, with no remainder.

The Factor Theorem

Ifis a factor of the polynomialthenhas no remainder or equivalently,

We show uses of the factor theorems in the following examples.

Example: Show thatis a factor of

Solveto get

thereforeis a factor of

Example

Find the polynomial with with integer coefficients whose roots are

If these are the roots, the factors areororandhence we expand and simplifyto get#

Example

Whenis divided byorthe remainder is zero. Find a andhence find the polynomial

Remainder 0 whenis divided by

Remainder 0 whenis divided by

Now we solve the simultaneous equations

(1)

(2)

hence