Question

Si (x-a)|f(x) y (x-b)|f(x) demostrar que (x-a)(x-b)|f(x)

Answer 1

$[1] ~~ (x-a)|f(x)\equiv f(x)=(x-a)P(x)\\\\\ [2] ~~ (x-b)|f(x)\equiv  (x-b)|(x-a)P(x)\equiv (x-b)|P(x) \\\\\text{Ya que a\neq b y (x-a) es un factor primo, por ende:}\\\\P(x)=(x-b)\cdot\Psi(x)\\\\\ [3]~~ f(x)=(x-a)(x-b)\cdot\Psi(x) \equiv (x-a)(x-b) | f(x) \\\\\hspace*{9cm} \square\square\square$