Giải thích các bước giải:
Ta có:
$Q(x)=(\dfrac{x^2}{2}-\dfrac12x^3+\dfrac12x)x-(\dfrac{x^5}{3}-\dfrac12x^4+x^2-\dfrac{x^5}{3})$
$\to Q(x)=(\dfrac{x^2}{2}-\dfrac12x^3+\dfrac12x)x-(\dfrac{x^5}{3}-\dfrac{x^5}{3}-\dfrac12x^4+x^2)$
$\to Q(x)=(\dfrac{x^2}{2}-\dfrac12x^3+\dfrac12x)x-(0-\dfrac12x^4+x^2)$
$\to Q(x)=(\dfrac{x^2}{2}-\dfrac12x^3+\dfrac12x)x-(-\dfrac12x^4+x^2)$
$\to Q(x)=\dfrac{x^3}{2}-\dfrac12x^4+\dfrac12x^2+\dfrac12x^4-x^2$
$\to Q(x)=-\dfrac{1}{2}x^4+\dfrac{1}{2}x^4+\dfrac{x^3}{2}+\dfrac{1}{2}x^2-x^2$
$\to Q(x)=\dfrac{x^3-x^2}{2}$
$\to Q(x)=\dfrac{x^2(x-1)}{2}$
Ta có $x\in Z\to x, x-1$ là $2$ số tự nhiên liên tiếp
$\to x(x-1)\quad\vdots\quad 2$
$\to x^2(x-1)\quad\vdots\quad 2$
$\to \dfrac{x^2(x-1)}{2}\in Z,\quad\forall x\in Z$
$\to đpcm$
