$$ \text{a,b,c,d,e }\in \text{R, and a}+\frac{\text{b}}{2}+\frac{\text{c}}{3}+\frac{\text{d}}{4}+\frac{\text{e}}{5}=0 \\ \text{Prove that a}+\text{bx}+\text{cx}^2+\text{dx}^3+\text{ex}^4=\text{0 has at least one real zero}. $$
The question is from The Art and Craft of Problem Solving. I can only observe that if we can prove there is an x such that f(x)*e<0, then we are done. But I can not use the first condition well.
Hint: Consider $F(x)=ax+\dfrac{b}{2}x^2+\dfrac{c}{3}x^3+\dfrac{d}{4}x^4+\dfrac{e}{5}x^5$ and $f(x)=a+bx+cx^2+dx^3+ex^4$.