I have a polynomial $p(x)=a_Nx^N+a_{N-1}x^{N-1}+\dots+a_0$. I want to convert this into another polynomial of same order, say $b_Ny^N+b_{N-1}y^{N-1}+\dots+b_0$. Is it possible to find a transformation $y=f(x)$ that will transform the coefficients $a_n$ into $b_n$? For the case of second order polynomial, a linear transform does the job.

I am asking this in order to generate random variables. For gaussian random variable case I can sample from $N(0,1)$, density of which is a second order polynomial inside an exponential. By a linear transformation $ax+b$, I can get samples from $N(b,a^2)$. What I am trying to do is assuming I have samples from $x \sim density\propto \exp(a_Nx^N+a_{N-1}x^{N-1}+\dots+a_0)$, whether or not I can get samples from $y \sim \exp(b_Ny^N+b_{N-1}y^{N-1}+\dots+b_0)$ just by a deterministic transform $y=f(x)$.

Btw, $a_N$ is a negative constant and $N$ is always even, so that $\exp(polynomial)$ makes a sensible probability density function. Thanks a lot in advance.

**edit**
As Robert Israel stated in his comment, the constant term is to be a proper normalization term. The basic question is this, a gaussian corresponds to a log-polynomial density where the polynomial is of 2nd order, and one can sample $N(\mu,\sigma^2)$ from $N(0,1)$ just by the transform $\mu+\sigma x$. Hence this corresponds to converting a polynomial into another one. Constant term is not of importance as it is just a normalization term. So for an higher order polynomial of degree N is there a possible way to do such a thing?