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        riddles >> medium >> Polynomial
        
(Message started by: cool_joh on Aug 18^th, 2008, 12:50am)

Title: Polynomial
Post by cool_joh on Aug 18^th, 2008, 12:50am

Find all monic integer polynomials p(x) of degree two for which there exists an integer polynomial q(x) such that p(x)q(x) is a polynomial having all
coeffcients ±1.

What if p(x)q(x) is allowed to have zero coefficients?

Title: Re: Polynomial
Post by Hippo on Aug 18^th, 2008, 2:06am

Equivalently: find a,b such that the sequence A₀=0,A₁=1,A_n+2=aA_n+1+bA_n±1 has only finite number of nonzero elements (for appropriate choice of ± signs).

Title: Re: Polynomial
Post by Hippo on Aug 18^th, 2008, 2:57am

So when A_n+1=A_n+2=0 we get bA_n=±1 therefore b=±1 and A_n=±1.

Hmmm, my choice of variabes p,q may be rather confusing with independent polynomial variables p(x),q(x) in the original question :( ... I've converted it to a,b.

Title: Re: Polynomial
Post by Eigenray on Aug 20^th, 2008, 4:39am

This one finally just hit me: [hide]the roots of p(x) can't be too big[/hide].

Title: Re: Polynomial
Post by Hippo on Aug 20^th, 2008, 1:12pm

on 08/20/08 at 04:39:27, Eigenray wrote:

This one finally just hit me: [hide]the roots of p(x) can't be too big[/hide].

Yes, this is much nicer solution ;) (at least my one was not finished).