Question

\(Q_{\left(x\right)}=\frac{1}{2}x^3-\frac{1}{2}x^2\)

CM Q thuộc Z với mọi X thuộc Z

Answer 1

\(Q\left(x\right)=\frac{1}{2}x^3-\frac{1}{2}x^2=\frac{1}{2}x.x.\left(x-1\right)\)

do \(x\in Z\Rightarrow\left[\begin{matrix}x=2n\left(1\right)\\x=2n+1\left(2\right)\end{matrix}\right.\) với \(n\in Z\)

TH1: \(x=2n\Rightarrow Q\left(x\right)=Q\left(n\right)=\frac{1}{2}.2n.2n\left(2n-1\right)=n^2\left(2n-1\right)\)

\(n\in Z\Rightarrow n^2.\left(2n-1\right)\in Z\Rightarrow dpcm\)(*)

TH2.

\(x=2n+1\Rightarrow Q\left(n\right)=\frac{1}{2}\left(2n+1\right)\left(2n+1\right)\left(2n+1-1\right)=\frac{1}{2}\left(2n+1\right)\left(2n+1\right).2n=n\left(2n+1\right)\left(2n+1\right)\)

\(n\in Z\Rightarrow n\left(2n+1\right)^2\in Z\Rightarrow dpcm\) (**)

(*) & (**) => dpcm