July 19th, 2017, 11:00 AM  #1 
Newbie Joined: Jul 2017 From: Denmark Posts: 11 Thanks: 0  p(x) for real x when p(p(x))+p(x)=x^4+3x^2+3
What I know: deg of p(x) is 2 due to that degree of deg g(x)+p(x)=highest deg of the equation. Hence p(x) can be written as ax^2+bx+c. But how do I continue? 
July 19th, 2017, 11:37 AM  #2 
Senior Member Joined: Oct 2009 Posts: 232 Thanks: 84 
If $p(x) = ax^2 + bx + c$, then what is $p(p(x))$?

July 19th, 2017, 12:13 PM  #3 
Newbie Joined: Jul 2017 From: Denmark Posts: 11 Thanks: 0 
p(p(x)) is then p(ax^2+bx+c) of course. Then we have p(ax^2+bx+c) + p(x) = x^4+3x^2+3. I have tried to go on with finding p(x), but I have three unknowns... 
July 19th, 2017, 12:17 PM  #4 
Senior Member Joined: Oct 2009 Posts: 232 Thanks: 84 
So can you simplify $p(ax^2 + bx + c)$??

July 19th, 2017, 12:20 PM  #5 
Newbie Joined: Jul 2017 From: Denmark Posts: 11 Thanks: 0 
Yes, into $\displaystyle p((x+m)(ax+n))+p(x)$. But do not understand how that is helping me?
Last edited by oscar3; July 19th, 2017 at 12:37 PM. 
July 19th, 2017, 12:23 PM  #6 
Senior Member Joined: Oct 2009 Posts: 232 Thanks: 84 
Why not simply say that $$p(p(x)) = a(p(x))^2 + bp(x) + c = a(ax^2 + bx + c)^2 + b(ax^2 + bx + c) + c$$ 
July 19th, 2017, 12:35 PM  #7 
Newbie Joined: Jul 2017 From: Denmark Posts: 11 Thanks: 0 
Smart! So now let's simplify: $\displaystyle a(ax^2+bx+c)^2+b(ax^2+bx+c)+c+(ax^2+bx+c)=a(ax^2+b x+c)^2+(b+1)(ax^2+bx+c)+c$ Or am I wrong? Last edited by skipjack; July 19th, 2017 at 09:45 PM. 
July 19th, 2017, 12:37 PM  #8 
Senior Member Joined: Oct 2009 Posts: 232 Thanks: 84  Right; this is $p(p(x)) + p(x)$.
Last edited by skipjack; July 19th, 2017 at 10:02 PM. 
July 19th, 2017, 12:38 PM  #9 
Newbie Joined: Jul 2017 From: Denmark Posts: 11 Thanks: 0 
But, should I not know the coefficients of it?
Last edited by skipjack; July 19th, 2017 at 09:35 PM. 
July 19th, 2017, 12:41 PM  #10 
Newbie Joined: Jul 2017 From: Denmark Posts: 11 Thanks: 0 
Never mind, it is not necessary


Tags 
3x2, algebra, polynomails, ppx, pxx4, real 
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