The roots of x^{3} + a x^{2} + b x + c = 0 are α, β and γ. Find the cubic whose roots are α^{3}, β^{3}, g^{3}.

**Solution**

x^{3} + (a^{3} - 3ab + 3c) x^{2} + (b^{3} - 3abc + 3c^{2})x + c^{3} = 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: α^{3} + β^{3} + γ^{3} etc. You obviously start with (α + β + γ)^{3} etc and then add additional terms to get the desired expressions.

© John Scholes

jscholes@kalva.demon.co.uk

4 Sep 1999