Algebra question I find hard to answer

Question

Let α, β, and γ be the roots of x³ − 4x − 8 = 0. Find the numerical value of the expression

[(α + 2) / (α − 2)] + [(β + 2) / (β − 2)] + [(γ + 2) + (γ − 2)]

Answer 1

This is a problem about elementary symmetric polynomials. If you expand out the expression (x-α)(x-β)(x-γ), you see that the x² term is -(α+β+γ), the x term is αβ+αγ+βγ, and the constant term is -αβγ. Since you know the coefficients of your polynomial, you know that

α+β+γ=0

αβ+αγ+βγ=-4

αβγ=8

These expressions are called the elementary symmetric polynomials in three variables. The problem is now to write your expression in terms of these polynomials. I assume your final terms is meant to be (γ + 2)/(γ − 2), else the problem cannot be solved this way.

It is always possible to rewrite a symmetric polynomial in terms of these elementary symmetric polynomials. Your expression is not a polynomial, but we can rewrite it as a quotient of two polynomials by finding a common denominator.

[(α+2)(β-2)(γ-2) + (α-2)(β+2)(γ-2) + (α-2)(β-2)(γ+2)]/ ((α-2)(β-2)(γ-2))

Expanding out the numerator and denominator takes some doing, but you should eventually get

[3αβγ-2αβ-2αγ-2βγ-4α-4β-4γ+24] / [αβγ-2αβ-2αγ-2βγ+4α+4β+4γ-8]

Now you can find the elementary symmetric polynomials hidden here:

[3αβγ-2(αβ+αγ+βγ)-4(α+β+γ)+24] / [αβγ-2(αβ+αγ+βγ)+4(α+β+γ)-8]

Plug in the known values, and simplify

[3·8-2·(-4)-4·0+24]/[8-2(-4)+4·0-8]

[24+8+24] / [8+8-8]

56/8

7

Answer 2

The three roots are (you choose who gets to be α):

2.6494, -1.3247 ± 1.1246i

Can you substitute, calculate. and answer?