Writing a polynomial as product of its roots

Question

A polynomial $$p=\sum_{i=0}^na_ix^i$$ with roots $x_i, i=1, \cdots, n$ is given. Is it true to say that for any value of $x$ $$\sum_{i=0}^na_ix^i=\prod_{i=1}^n(x-x_i)$$ Is this true?

Answer 1

It's almost true. You just need to tack the leading coefficient onto the product: $$\sum_{i=0}^na_ix^i=a_n\prod_{i=1}^n(x-x_i)$$