1. ## Finding Polynomial

Suppose $P(x)$ is a polynomial which satisfies the following condition:

$P(P'(x)) = 27x^{6}-27x^{4}+6x^{2}+2.$

find a possible polynomial, P(x), that satisfies the above condition.

2. ## Re: Finding Polynomial

solve

<a href="http://www.codecogs.com/eqnedit.php?latex=27{&space;x&space;}^{&space;6&sp ace;}-27{&space;x&space;}^{&space;4&space;}&plus;6{&spac e;x&space;}^{&space;2&space;}&plus;2=\left(&space; 3x^{&space;2&space;}&plus;2ax&plus;b&space;\right) &space;^{&space;3&space;}&plus;a\left(&space;3x^{& space;2&space;}&plus;2ax&plus;b&space;\right)&spac e;^{&space;2&space;}&plus;b\left(&space;3x^{&space ;2&space;}&plus;2ax&plus;b&space;\right)&space;&pl us;c" target="_blank"><img src="http://latex.codecogs.com/gif.latex?27{&space;x&space;}^{&space;6&space;}-27{&space;x&space;}^{&space;4&space;}&plus;6{&spac e;x&space;}^{&space;2&space;}&plus;2=\left(&space; 3x^{&space;2&space;}&plus;2ax&plus;b&space;\right) &space;^{&space;3&space;}&plus;a\left(&space;3x^{& space;2&space;}&plus;2ax&plus;b&space;\right)&spac e;^{&space;2&space;}&plus;b\left(&space;3x^{&space ;2&space;}&plus;2ax&plus;b&space;\right)&space;&pl us;c" title="27{ x }^{ 6 }-27{ x }^{ 4 }+6{ x }^{ 2 }+2=\left( 3x^{ 2 }+2ax+b \right) ^{ 3 }+a\left( 3x^{ 2 }+2ax+b \right) ^{ 2 }+b\left( 3x^{ 2 }+2ax+b \right) +c" /></a>

for P(x)=x^3+ax^2+bx+c
by inspection or otherwise, one can deduce that a=0, b=-1, c=2 suffices the equation
therefore P(x)=x^3-x+2

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