If $u = ax^2+a_2x+a_3$ is a quadratic polynomial then there exists a solution to $w=v^2$ where $w=cu+c_2$ and $v = bx+b_2$, which can be easily shown to be true.

For instance when $u=x^2+x+1$, $w=4u-3$, and $v=2x+1$.

What if $u=ax^3+a_2x^2+a_3x+a_4$, $v=bx^2+b_2x+b_3$, and $w=cu^2+c_2u+c_3$? Is there always a solution to $w=v^3$? It is true in some cases, but seems uncertain in others.

What about the more general question where $u$ is a degree $n$ polynomial, $v$ is a degree $n-1$ polynomial and $w$ is a degree $n-1$ polynomial in terms of $u$? Does there always exist a solution to $w=v^n$?


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