Problem: (360 Problems For Mathematical Contests – 1.1.53) Let
be a polynomial with complex coefficients such that there is an integer with
Prove that the polynomial has at least a zero with absolute value less than .
Solution: We will prove the result by contradiction. Assume all the zeros have absolute value (modulus) at least . Let the zeros be . Our assumption says that for all .
By Vieta’s Formulas, we have
where the summation is taken over all sets of roots.
Then, by the Triangle Inequality for complex numbers,
But since for all by our assumption, we know
and similarly for all other sets of roots. Since there are precisely sets of roots, we have
Connecting the inequalities, we find that
giving us a contradiction. Hence our original assumption is false and their exists a root such that as desired. QED.
Comment: Once again, Vieta’s Formulas are extremely important to know. Also, being able to manipulate the norms of complex numbers and knowing the general properties of them is essential to solving this problem.
Practice Problem: (360 Problems For Mathematical Contests – 1.1.58) Consider the equation
with real coefficients . Prove that if the equation has all of its roots real, then . Is the reciprocal true?