# Finding polynomials sattisfying P(−c+K/(u+c))(u+c)2/K=P(u)

Finding polynomials sattisfying $P\left(-c+K/\left(u+c\right)\right)\left(u+c{\right)}^{2}/K=P\left(u\right)$
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esbalatzaj
When u tends to infinity then the modulus of the left side of the equality (*) grows not faster than a polynomial of degree 2. Therefore the polynomial from the right side of the equality has a degree at most two. Now, assuming that $P\left(x\right)=A{x}^{2}+Bx+C$ and comparing coefficients from both sides, we obtain a system
$\left\{\begin{array}{l}A=\frac{1}{K}\left(A{c}^{2}-Bc+C\right)\\ B=\frac{2c}{K}\left(A{c}^{2}-Bc+C\right)-2Ac+B\\ C=\frac{{c}^{2}}{K}\left(A{c}^{2}-Bc+C\right)+c\left(-2Ac+B\right)\end{array}$
Solving it, we obtain that $P\left(u\right)\equiv B\left(u+c\right)$ or a degenerated solution K=0 and P(−c)=0.