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2 . \(\dfrac{n^7+n^2+1}{n^8+n+1}=\dfrac{n^7-n+n^2+n+1}{n^8-n^2+n^2+n+1}\)
\(=\dfrac{n\left(n^6-1\right)+n^2+n+1}{n^2\left(n^6-1\right)+n^2+n+1}=\dfrac{n\left(n^3+1\right)\left(n^3-1\right)+n^2+n+1}{n^2\left(n^3+1\right)\left(n^3-1\right)+n^2+n+1}\)\(=\dfrac{n\left(n^3+1\right)\left(n-1\right)\left(n^2+n+1\right)+n^2+n+1}{n^2\left(n^3+1\right)\left(n-1\right)\left(n^2+n+1\right)+n^2+n+1}\)
\(=\dfrac{\left(n^2+n+1\right)\left[\left(n^4+n\right)\left(n-1\right)\right]}{\left(n^2+n+1\right)\left[\left(n^5+n^2\right)\left(n-1\right)+1\right]}\)
\(=\dfrac{n^5-n^4+n^2-n}{n^6-n^5+n^3-n^2+1}=\dfrac{n^4\left(n-1\right)+n\left(n-1\right)}{n^5\left(n-1\right)+n^2\left(n-1\right)+1}\)
\(=\dfrac{\left(n-1\right)\left(n^4+n\right)}{\left(n-1\right)\left(n^5+n^2\right)+1}\)
Vậy ,với mọi số nguyên dương n thì phân thức trên sẽ không tối giản
\(\frac{n^7+n^2+1}{n^8+n+1}=\frac{\left(n^2+n+1\right)\left(n^5-n^4+n^2-n+1\right)}{\left(n^2+n+1\right)\left(n^6-n^5+n^3-n^2+1\right)}=\frac{n^5-n^4+n^2-n+1}{n^6-n^5+n^3-n^2+1}\)
=>phân số ban đầu chưa tối giản với mọi n
Ta có :
\(\frac{n^7+n^2+1}{n^8+n+1}=\frac{n^7-n^4+n^4-n+n^2+n+1}{n^8-n^5+n^5-n^2+n^2+n+1}\)
\(=\frac{n^4\left(n^3-1\right)+n\left(n^3-1\right)+\left(n^2+n+1\right)}{n^5\left(n^3-1\right)+n^2\left(n^3-1\right)+\left(n^2+n+1\right)}\)
\(=\frac{n^4\left(n-1\right)\left(n^2+n+1\right)+n\left(n-1\right)\left(n^2+n+1\right)+\left(n^2+n+1\right)}{n^5\left(n-1\right)\left(n^2+n+1\right)+n^2\left(n-1\right)\left(n^2+n+1\right)+\left(n^2+n+1\right)}\)
\(=\frac{\left(n^2+n+1\right)\left(n^5-n^4+n^2-n+1\right)}{\left(n^2+n+1\right)\left(n^6-n^5+n^3-n+1\right)}\)
\(=\frac{n^5-n^4+n^2-n+1}{n^6-n^5+n^3-n+1}\)
Do phân số \(\frac{n^7+n^2+1}{n^8+n+1}\) còn thu gọi được thành \(\frac{n^5-n^4+n^2-n+1}{n^6-n^5+n^3-n+1}\) nên nó chưa tối giản (đpcm)
Gọi \(d=\left(n^3+2n;n^4+3n^2+1\right)\)
\(\Rightarrow\hept{\begin{cases}\left(n^3+2n\right)⋮d\\\left(n^4+3n^2+1\right)⋮d\end{cases}}\Leftrightarrow\hept{\begin{cases}n\left(n^3+2n\right)=\left(n^4+2n^2\right)⋮d\\\left(n^4+3n^2+1\right)⋮d\end{cases}}\)
\(\Rightarrow\left(n^4+3n^2+1\right)-\left(n^4+2n^2\right)⋮d\)
\(\Leftrightarrow n^2+1⋮d\Leftrightarrow\left(n^2+1\right)^2⋮d\)
\(\Rightarrow\left(n^2+1\right)^2-\left(n^4+2n^2\right)⋮d\Leftrightarrow1⋮d\Rightarrow d=1\)
=> P/s tối giản
Gọi \(d=ƯCLN\left(n^3+2n;n^4+3n^2+1\right);\left(d>0\right)\)
\(\Rightarrow\hept{\begin{cases}n^3+2n⋮d\left(1\right)\\n^4+3n^2+1⋮d\end{cases}}\)
Từ \(\left(1\right)\): \(\Rightarrow n\left(n^3+2n\right)⋮d\)
\(\Rightarrow n^4+2n^2⋮d\)
\(\Rightarrow\left(n^4+3n^2+1\right)-\left(n^4+2n^2\right)⋮d\)
\(\Rightarrow n^2+1⋮d\)
\(\Rightarrow\left(n^2+1\right)^2⋮d\)
\(\Rightarrow n^4+2n^2+1⋮d\)
\(\Rightarrow1⋮d\)(do \(n^4+2n^2⋮d\))
Vì \(d>0\)\(\Rightarrow d=1\)
\(\Rightarrow\left(n^3+2n;n^4+3n^2+1\right)=1\)
\(\Rightarrow\frac{n^3+2n}{n^4+3n^2+1}\)là phân số tối tối giản với mọi n nguyên