roots of polynomials

Question

Caon someone help me understand how to calculate the roots of polynomials?

Here is the question.

Let α, β, γ, and δ be roots of the polynomial equation:

x^4 + 5x^3 + 3x^2 – x + 2 = 0

Calculate the exact values of:

α^2 + β^2 + γ^2 + δ^2;

α^-1 + β^-1 + γ-1 + δ^-1;

α^3 + β^3 + γ^3 + δ^3.

Answer 1

Given polynomial equation is x^4+5x^3+3x^2-x+2 = 0

It s a fourth degree equation ax^4+bx^3+cx^2+dx+e = 0

a = 1, b = 5, c = 3, d = -1, e = 2

Now the roots of this equation are α, β, γ,δ then

α+β+ γ+δ = -b/a

αβ+αγ+αδ+δα+βγ+βδ+γδ = c/a

αβγ+αβδ+αγδ+βγδ = -d/a

αβγδ = e/a

α+β+ γ+δ = -b/a --> -5

αβ+αγ+αδ+δα+βγ+βδ+γδ = c/a --> 3

αβγ+αβδ+αγδ+βγδ = -d/a --> 1

αβγδ = e/a --> 2

1) α^2 + β^2 + γ^2 + δ^2 =

We know that (a+b)^2 = a^2+b^2+2ab

a^2+b^2 = -(a+b)^2-2ab

Similarly α^2 + β^2 + γ^2 + δ^2 =(α+β+ γ+δ )^2 -2(αβ+αγ+αδ+δα+βγ+βδ+γδ)

= 25-2(3) = 25-6 = 19

α^2 + β^2 + γ^2 + δ^2 = 19