Homework 1

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Homework

P. 1.1. Using the identity (u + v)3 − 3uv(u + v) − u3 − v 3 = 0 , solve the reduced 3rd degree equation x3 + px + q = 0 , by performing the substitution x = α2 β + αβ 2 and obtaining a 2nd degree equation whose roots are α3 and β 3 . P. 1.2. Solve the 4th degree equation x4 + ax3 + bx2 + cx + d = 0 by performing the substitution y = x + a4 , to obtain a reduced equation y 4 + py 2 + qy + r = 0. By coefficient identification in y 4 + py 2 + qy + r = (y 2 + αy + β)(y 2 − αy + γ) , express β and γ in terms of α and obtain a 3rd degree equation with respect to α2 . One solution of this resolvent equation leads then to a factorization which reduces the solution of the reduced equation to the solution of two 2nd degree equations y 2 + αy + β = 0

and y 2 − αy + γ = 0 .

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