The roots of x3 + a x2 + b x + c = 0 are α, β and γ. Find the cubic whose roots are α3, β3, g3.
x3 + (a3 - 3ab + 3c) x2 + (b3 - 3abc + 3c2)x + c3 = 0.
A routine manipulation. Suppose the roots are α, β, γ. Then α + β + γ = - a, αβ + βγ + γα = b, αβγ = -c. So to get the coefficients of the desired polynomial we have to find the corresponding expressions in the cubes: α3 + β3 + γ3 etc. You obviously start with (α + β + γ)3 etc and then add additional terms to get the desired expressions.
2nd Putnam 1939
© John Scholes
4 Sep 1999