Functions which permute the roots of an equation

Extract

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.