A-LEVEL MATHEMATICS Topic : The Factor Theorem

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