tìm x thuộc z để f(x) chia hết cho g(x) biết f(x)=x^3+2x^2+15, g(x)=x+3

Giải thích các bước giải:

 Ta có:

\(\begin{array}{l}
f\left( x \right) = {x^3} + 2{x^2} + 15\\
 = \left( {{x^3} + 3{x^2}} \right) – \left( {{x^2} + 3x} \right) + \left( {3x + 9} \right) + 6\\
 = {x^2}\left( {x + 3} \right) – x\left( {x + 3} \right) + 3\left( {x + 3} \right) + 6\\
 = \left( {x + 3} \right)\left( {{x^2} – x + 3} \right) + 6\\
 = g\left( x \right).\left( {{x^2} – x + 3} \right) + 6\\
f\left( x \right) \vdots g\left( x \right) \Rightarrow 6 \vdots g\left( x \right) \Leftrightarrow 6 \vdots \left( {x + 3} \right)\\
x \in Z \Rightarrow \left( {x + 3} \right) \in Z \Rightarrow x + 3 \in \left\{ { – 6; – 3; – 2; – 1;1;2;3;6} \right\}\\
 \Rightarrow x \in \left\{ { – 9. – 6; – 5; – 4; – 2; – 1;0;3} \right\}
\end{array}\)