Application of Rolle's Theorem

Let $a_0,a_1,a_2$ and $a_3$ be real numbers such that $a_0+\frac{a_1}{2}+\frac{a_2}{3}+\frac{a_3}{4}=0$. Then show that the polynomial $f(x)=a_0+a_1x+a_2x^2+a_3x^3$ has at least one root in the interval $( 0 , 1 )$. Consider the polynomial $g(x) = a_0x+\frac{a_1}{2}x^2+\frac{a_2}{3}x^3+\frac{a_3}{4}x^4$ Clearly $g(0)=0$ and $g(1) = a_0+\frac{a_1}{2}+\frac{a_2}{3}+\frac{a_3}{4} = 0$ { given in the problem } Since $g(x)$ is a polynomial it is continuous in [0,1] and differentiable in (0,1) Thus by Rolle's Theorem, $g'(x) = 0$ for at least one $x \in (0,1)$ $\Rightarrow a_0+a_1x+a_2x^2+a_3x^3 = 0$ for at least one $x \in (0,1)$