An amusing exercise to give to a mathematical sixth form is:

If the roots of the equation

F1(x) ≡ x3 + x2 − 2x − 1 = 0 (1)

are α, β, γ, form the equation whose roots are α2 − 2, β2 − 2, γ2 − 2.

The usual methods used will be, either to find the coefficients of the new equation by forming the appropriate symmetric functions of α, β, γ; or else to eliminate x from the relation y = x2 – 2 and the given cubic equation.