If
is a factor of
then
so
is a root ofor equivalently, a solution of the equation
Example: Show that
is a factor of
henceis a factor of
Example:
and
are both roots of the quadratic expression
Find
Sinceandare both roots
and
are both factors. A quadratic equation has at most two real factors/roots hence
More complicated questions may involve simultaneous equations:
and
are 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 of
and by considering the constant term 1