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Find polynomial with irrational root and solve an equation.

Source: Vietnam MO 1984 P1

March 20, 2011

algebrapolynomialcalculusintegrationalgebra unsolved

Problem Statement

(a)

Find a polynomial with integer coefficients of the smallest degree having

\sqrt{2} + \sqrt[3]{3}

as a root.

(b)

Solve 1 +\sqrt{1 + x^2}(\sqrt{(1 + x)^3}-\sqrt{(1- x)^3}) = 2\sqrt{1 - x^2}.