Is this a correct interpretation of the fundamental theorem of algebra? I tried reading the Wikipedia page but it's stated in terms of complex roots, and I don't really understand how that relates to the following proposition:
if a real valued polynomial:
$$\sum_{i=0}^n a_i x^i = 0$$
for all $x \in \mathbb{R}$, is it right to say the Fundamental Theorem of algebra implies that $a_i = 0$ for all $i$?
This was stated off handedly during a recent talk and I am not sure if I heard it correctly, since when I look up the theorem it is difficult for me to understand the way that page is written

There are two questions in your post. The answers to them are different.

Is this a correction interpretation of the fundamental theorem of algebra?

No, according to the standard meaning of "fundamental theorem of algebra," which says every polynomial with real coefficients has at least one complex root. That theorem is different and harder than the theorem you need for the rest of your post.

Is it right to say the Fundamental Theorem of algebra implies that $a_i=0$ for all $i$?

Yes, the fundamental theorem implies your result. But your result follows from a weaker theorem that holds over fields that aren't necessarily algebraically closed: every polynomial of degree $n\geq 1$ has at most $n$ roots.
