Okay, this is frustrating. I can't seem to prove this elementary result.

Let $\displaystyle \lambda_1,\cdots,\lambda_k\in\mathbb{C}$ be distinct, i.e. $\displaystyle \lambda_i\neq\lambda_j$ for $\displaystyle i\neq j$. Also, let $\displaystyle a_1,\cdots,a_k\in\mathbb{C}\setminus\{0\}$, i.e. complex nonzero values.

Prove that $\displaystyle \sum_{i=1}^ka_i\prod_{\substack{j=1\\j\neq i}}^k(\lambda_j-X)$ is a nonzero polynomial in $\displaystyle X$.

I thought about using induction, but I can't see how to make that work. Directly expanding the coefficients is a nightmare, and one which doesn't seem to go anywhere either. I thought maybe there might be some linear algebra theory which would work, but I don't know. Any help would be much appreciated. Thanks!