Problem:
Prove that any monic polynomial (a polynomial with leading coefficient ) of degree with real coefficients is the average of two monic polynomials of degree with real roots.
Solution:
Solution: Let be monic real polynomial of degree . If . then for some real number . It is easy to see that is the average of and . wh of which has 1 real root. Now we assume that . Let polynomial
The degree of is . Consider the polynomials
We will show that for large enough these two polynomials have real roots. Since they are monic and their average is clearly , this will solve the problem.
Consider the values of polynomial at points . These values alternate in sign and are at least 1 (since at most two of the factors have magnitude 1 and the others have magnitude at least 2). On the other hand, there is a constant such that for , we have and . Take . Then we see that and evaluated at points alternate in sign. Thus polynomials and each has at least real roots. How ever since they are polynomials of degree , they must then each have real roots, as desired.
The problems on this page are the property of the MAA's American Mathematics Competitions