(In reply to
I would say by Kenny M)
I will start with P(x)*(x^40)-1=0. If this equation has 30 roots, then it must have at least 29 relative extremes, with one at least extreme occurring between each pair of consecutive roots.
The x-coordinates of those extremes occur at the roots of the derivative.
d/dx (P(x)*(x^40)-1)
= 40*P(x)*(x^39) + P'(x)*(x^40)
= x^39 * [40*P(x) + P'(x)*x]
40*P(x) + P'(x)*x is a polynomial of degree 20, so it has at most 20 roots. x^39 has only one root, x=0. So then the derivative has at most 21 roots. Not enough for the needed 29 roots that the derivative requires. So the answer to the original question is "No".
re: I would say - My solution
