\(A=\frac{B}{C}=\frac{n^5+n+1}{n^4+n^2+1}\)
\(B=n^5+n+1=\left(n^5+n^4+n^3\right)-\left(n^4+n^3+n^2\right)+\left(n^2+n+1\right)\)
\(B=n^3\left(n^2+n+1\right)-n^2\left(n^2+n+1\right)+\left(n^2+n+1\right)=\left(n^2+n+1\right)\left(n^3-n^2+1\right)\)
\(C=n^4+n^2+1=\left(n^4+n^3+n^2\right)-\left(n^3+n^2+n\right)+\left(n^2+n+1\right)\)\(C=n^2\left(n^2+n+1\right)-n\left(n^2+n+1\right)+\left(n^2+n+1\right)=\left(n^2+n+1\right)\left(n^2-n+1\right)\)
\(A=\frac{B}{C}=\frac{\left(n^2+n+1\right)\left(n^3-n^2+1\right)}{\left(n^2+n+1\right)\left(n^2-n+1\right)}\)
Ta có \(\left(n^2+n+1\right)\ne1\)với mọi n Thuộc N
=> với mọi n thuộc N phân số B và C luôn có ước chung là : \(\left(n^2+n+1\right)\ne1\) =>B/C không tối giản=> dpcm
