If $a_1$, $a_2$, $a_3$,$\cdots$ $a_n$ $(n\ge2)$ are real and $(n-1){a_1}^2-2na_2<0$ then prove that at least two roots of the equation $$f(x)=x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_n=0$$ are imaginary.
My work:
Let us prove this by strong induction. So we can clearly see that when we put $n=2$ we get $f(x)$ as a quadratic polynomial whose $\triangle<0$ therefore it has two imaginary roots. Therefore base case is true. Let the condition be true for all values from $2$ to $k$. Now let us prove this for $k+1$. Also let $$f(k)=x^k+a_1x^{k-1}+a_2x^{k-2}+\cdots+a_k=0$$ Now $$f(k+1)=x^{k+1}+a_1x^{k}+a_2x^{k-1}+\cdots+a_kx+a_{k+1}=0$$ Taking $x$ common from the whole equation we get $$f(k+1)=x\cdot f(k)+a_{k+1}=0$$ After this I am stuck. But after showing this to my teacher he said that this question will be done by using Rolle's theorm. I really don't know what Rolle's theorm is but I do know that it deals with continuity of functions. How to do it with Rolle's theorm$?$ And is strong induction helpful here$?$