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

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.

closed as off-topic by Kavi Rama Murthy, Cesareo, Jyrki Lahtonen, José Carlos Santos, Riccardo.AlestraMar 20 at 13:30

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• Do you know how to evaluate $(f+g)(u)$? – Randall Mar 20 at 2:24

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.