Problem:
There are real numbers , and such that is a root of and is a root of . These two polynomials share a complex root , where and are positive integers and . Find .
Solution:
Answer (330):
Because the given polynomials have real coefficients, the fact that they share a nonreal complex root implies that they share a pair of complex conjugate roots, and each polynomial is divisible by some quadratic polynomial , where and are real numbers. Thus
and
The coefficient of the quadratic term in the first polynomial is , so . The coefficient of the linear term in the second polynomial is , so . Therefore the two polynomials share the roots of , which, by the quadratic formula, are . The requested sum is .
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions.