Polynomial roots & Mathematical induction

[ben9703]

hi i have this homework question and I am 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>3my attempt:
-------------

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 I am 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 :)

 

[Dick]

hi i have this homework question and I am 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:
-------------

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 I am 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 :)

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.

 

[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

 

[Dick]

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

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.

 

[ben9703]

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