Let α, β, γ and δ denote the zeros of the quartic polynomial, f, given by

x^4 + 5x^3 − x^2 + 3x + 1.

i) Write down the four expressions for the elementary symmetric functions in α, β, γ and δ and give their values for this quartic polynomial.

ii) Express the symmetric functions α^2 + β^2 + γ^2 + δ^2 and Σ(α − β)^2

(six terms) in terms of these elementary symmetric functions and hence calculate their exact values. Calculate the exact value of α^(−3) + β^(−3) + γ^(−3) + δ^(−3).

So I know that the elementary functions are as follows:

Σα = α + β + γ + δ = -5

Σαβ = αβ + αγ + αδ + βγ + βδ + γδ = -1

Σαβγ = αβγ + αβδ + αγδ + βγδ = -3

αβγδ = 1

How do i proceed to do part (ii)? I am confused and the way my professor did it confused me even more. So can someone explain in detail?

Thanks in advance!