Sum Of Powers Of Roots Of Polynomials – Mattia Giuri's bizarre blog

Our olympic course professor gave us some exercises, and among these there is one which resolution is particularly instructive.

Find the sum of the fifth powers of the roots of:

$x^3-2x^2+5x+1 = 0$

To solve it, we need to write the recursion which is associated to the polynomial, which is:

$a_{n+3} = 2a_{n+2}-5a_{n+1}-a_n$

This recursion will give us the sum of the powers of the roots if we provide it with the right initial steps.

Let $\lambda_1, \lambda_2, \lambda_3$ be the roots of the polynomial, then:

$a_0 = \lambda_1^0 + \lambda_2^0+\lambda_3^0 = 3$

Then, by Vieta’s Theorem:

$a_1 = \lambda_1^1 + \lambda_2^1+\lambda_3^1 = 2$

Finally, also by Vieta’s Theorem, we have:

$a_2 = \lambda_1^2 + \lambda_2^2+\lambda_3^2 = a_1^2 - 2*5 = -6$

Now, by performing calculations with the recursion, we find

$a_5 = 87$