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2010 CHMMC Winter Tiebreaker 1 - f(f(\sqrt5 -\sqrt2)) if f(\sqrt5 +\sqrt2)= 0

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March 1, 2024
algebrapolynomialmonicCHMMC

Problem Statement

The monic polynomial ff has rational coefficients and is irreducible over the rational numbers. If f(5+2)=0f(\sqrt5 +\sqrt2)= 0, compute f(f(5−2))f(f(\sqrt5 -\sqrt2)). (A polynomial is monic if its leading coeffi cient is 11. A polynomial is irreducible over the rational numbers if it cannot be expressed as a product of two polynomials with rational coefficients of positive degree. For example, x2−2x^2 - 2 is irreducible, but x2−1=(x+1)(x−1)x^2 - 1 = (x + 1)(x - 1) is not.)
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