My Math Forum p(x) for real x when p(p(x))+p(x)=x^4+3x^2+3

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 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: 142 Thanks: 60 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: 142 Thanks: 60 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: 142 Thanks: 60 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
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Quote:
 Originally Posted by oscar3 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?
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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