The roots of a polynomial 
where
are constants are given by the solutions to
. Suppose though that we have the roots and want to find the polynomial. We can form the polynomialexpand the brackets and factorise.
Example: Find the polynomial with roots 1, 3, 5.
Often the roots are surds. Still we can follow the above procedure. The algebra is much simpler if we deal with the surds first.
Example: Find the polynomial with roots
and 4.
The above method only works if the coefficient of the highest power of
is 1. We usually have to present the answers in a form where the coefficients are all integers. Suppose we are told the roots are
The expression
has these as roots. Because of the fraction1 over 2, we should now multiple by 2 to get
then expand the brackets to get