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P(x) = (x - u^k) (x - uv) (x -v^k) = x^3 + ax^2 + bx + c

Source: Austrian - Polish 1992 APMC

May 7, 2020
polynomialalgebra

Problem Statement

Let kk be a positive integer and u,vu, v be real numbers. Consider P(x)=(x−uk)(x−uv)(x−vk)=x3+ax2+bx+cP(x) = (x - u^k) (x - uv) (x -v^k) = x^3 + ax^2 + bx + c. (a) For k=2k = 2 prove that if a,b,ca, b, c are rational then so is uvuv. (b) Is that also true for k=3k = 3?
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