# Thread: Elementary Functions and Polynomials

1. ## Elementary Functions and Polynomials

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?

2. $a^2+b^2+c^2+d^2 = (a+b+c+d)^2-2(ab+ac+bc+ad+bd+cd).$

3. $(a-b)^2 + (a-c)^2 + (a-d)^2 + (b-c)^2 + (b-d)^2 + (c-d)^2$

$= a^2 - 2ab + b^2 + a^2 - 2ac + d^2 + a^2 - 2ad + d^2 + b^2 - 2bc$

$+ c^2 + b^2 - 2bd + d^2 + c^2 - 2cd + d^2$

$= 3(a^2 + b^2 + c^2 + d^2) - 2(ab + ac + ad + bc + bd + cd)$

now use TheCoffeeMachine's result above to finish.