Polynomial roots & Mathematical induction

1. Jan 23, 2014

ben9703

hi i have this homework question and im not sure if my thought process is valid.

The Question:

let a, b and c be roots of the polynomial equation: x^3+px+q=0 and S(n)=(a^n)+(b^n)+(c^n)

now prove: that for S(n)= -p(S(n-2))-q(S(n-3)) for n>3

my attempt:
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first off i just played around with the equation. S(1)=0 , S(2)=-2p , S(3)=-3q after this all following values can be found by rearranging the polynomial to... x^3=-px-q and multiplying through x for each successive value of n. from this its clear how the above function ( S(n)= -p(S(n-2))-q(S(n-3)) ) arises. what im not sure about is the pure mathematical proof to proving this (n>3).

if anybody could shine some light on the question or just share ideas i would be very grateful. Thanks :)

2. Jan 23, 2014

Dick

I think you are already done. $x^n=-px^{n-2}-qx^{n-3}$, by multiplying your relation through by powers of x. So summing over a,b,c gives you S(n)=-p(S(n-2))-q(S(n-3)). I'm not even sure you needed to find those initial values, unless you actually want to calculate the values of S(n) using the recursion.

3. Jan 23, 2014

ben9703

Yeah that's what I was thinking but I'm still not sure how to prove the domain n>3 . thanks for the reply

4. Jan 23, 2014

Dick

I don't think you need to worry about that. The problem just says to show it works if n>3. It doesn't say you have to show that it doesn't work for n<=3. Though if p=q=0 you might have a problem with that case.

5. Jan 24, 2014

ben9703

Ah yeah I think your right. Thanks for answering when no one else did :)