Question: Given a nonconstant polynomial P with at least one negative coefficient, then $(P(x))^m\forall m\in N^{+},m\ge 2$ with all coefficient is postive.
in other words:
Prove :There exsit real coefficient polynomial $$P(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0},n\ge 1$$ where there exsit some coefficient is negative.
and
such $$(P(x))^m=b_{nm}x^{nm}+\cdots+b_{1}x+b_{0},\forall m\in N^{+},m\ge 2$$where $b_{i}>0,i=1,\cdots,nm$
I fell maybe this polynomial is not exsit,But I can't prove it? Thank you
My try:if let $$P(x)=x-1,\Longrightarrow (P(x))^2=x^2-2x+1,(P(x))^3=x^3-3x^2+3x+1,\cdots$$ this example is not such this condition
if $$P(x)=x^2-x+1,\Longrightarrow (P(x))^2=x^4-2x^3+3x^2-2x+1$$ also not such condition
