Question

Dati un exemplu de 3 numere intregi a , b , c astfel incat sa aiba loc egalitatea: x³-3x²-10x=(x+a)·(x+b)·(x+c) pt orice numar real x.

Answer 1

   
[tex]\displaystyle\\ \text{Se da: }\\ x^3-3x^2-10x=(x+a)(x+b)(x+c)\\ \text{Se cere" }\\ a, ~b, ~c \in Z\\ \text{Practic se cere descompunerea polinomului }~ x^3-3x^2-10x\\\\ \text{Rezolvam }\\ x^3-3x^2-10x=\\ =x(x^2-3x-10)=\\ ~~~~~~~~(-3x = -5x+2x)\\ =x(x^2-5x+2x-10)=\\ =x(x(x-5)+2(x-5))=\\ =\boxed{x(x-5)(x+2)}\\ ~~~~~~~~\text{Il scriem sub forma: }~(x+a)(x+b)(x+c)\\\\ x(x-5)(x+2) = \boxed{(x+0)(x+(-5))(x+2)}\\\\ \boxed{\Longrightarrow~~\begin{cases} a=0\\b=-5\\c=2 \end{cases}}[/tex]