Finding a Polynomial With Given Roots
The roots of a polynomial whereare 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 rootsand 4.
The above method only works if the coefficient of the highest power ofis 1. We usually have to present the answers in a form where the coefficients are all integers. Suppose we are told the roots areThe expressionhas these as roots. Because of the fraction1 over 2, we should now multiple by 2 to getthen expand the brackets to get