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.Alestra Mar 20 at 13:30
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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.