wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> medium >> A polynomial
        
(Message started by: wonderful on Jul 2^nd, 2008, 9:28pm)

Title: A polynomial
Post by wonderful on Jul 2^nd, 2008, 9:28pm

Find a polynomial P(x) of degree n ( x in R) which has real root(s) and P(x).P(2x^2)=P(2x^3+x) for every x in R

Have A Great Day!

Title: Re: A polynomial
Post by Eigenray on Jul 2^nd, 2008, 11:09pm

Do you mean it has a real root, or that all roots are real?

If the latter, [hide]there is no such polynomial, because if x is a non-zero real root, then 2x³+x is another root of larger norm[/hide].

And if the former, [hide]there is also no such polynomial, because the only possible real root is x=0, and then we can show inductively that all coefficients must be 0[/hide].

Title: Re: A polynomial
Post by wonderful on Jul 2^nd, 2008, 11:40pm

For the moment, we can assume that all roots of P(x) are real. I think you forgot the term P(x) in the product P(x).P(2x^2+2), didn't you?

Have A Great Day!

Title: Re: A polynomial
Post by Eigenray on Jul 2^nd, 2008, 11:58pm

~~No, I think you forgot the term +2 in P(x).P(2x²+2). :P~~

If [hide]x is a root of P, then 2x³+x is also a root. Since there are only finitely many roots, if x is real it must be 0[/hide].

So we can't have every root real.