If two polynomials share a root, does their sum/product also share that root?

Question

If we have two polynomials $f$ and $g$ such that $f(u) = 0$ and $g(u) = 0$, is it true that $(f+g)(u) = 0$ and/or $(f\cdot g)(u) = 0$? My intuition leads me to believe this is true, but I feel like I am missing something.

Answer 1

If $f(u) = g(u) = 0$ then, by definition, $$ (fg)(u) = f(u)g(u) = 0 \cdot 0 = 0. $$ Similarly, $$ (f+g)(u) = f(u)+g(u) = 0 +0 = 0. $$ Note that $f$ and $g$ need not be polynomials for this reasoning to apply.