# Solutions to cubic and higher degree functions

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 $$w=v^n$$?