Firstly, just getting back into maths after a very long time, so please accept my apologies if this is too basic.
Given a polynomial $y^5=3x^4+4x^3+5x^2+6x+7$, it seems clear that $y^5$ is a f(x). Does this mean that I can use Descartes Rule of Signs to rule out the possibility of any positive solutions for x or do I have to find a way to provide a concrete solution for the general equation first?
Many thanks. Rob
Descarte's rule of signs is about single-variable polynomials. It says that the polynomial on the right side can't be zero for positive $x$. (So you're right about that.) You can conclude that there is no positive $x$ that makes $y^5 = 0$. Hence, (if we're working in the real numbers) $y$ must be positive for positive $x$.
But note, that if you had an even power of $y$, you could have $y^4=16$ and then you could still have valid negative values for $y$.
