Gọi \(d=ƯCLN\left(n^3+2n;n^4+3n^2+1\right)\) \(\left(d\in N\right)\)
\(\Rightarrow\left\{{}\begin{matrix}n^3+2n⋮d\\n^4+3n^2+1⋮d\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}n^4+2n^2⋮d\\n^4+3n^2+1⋮d\end{matrix}\right.\)
\(\Rightarrow\) \(n^2+1⋮d\)
Mà \(n^3+2n⋮d\)
\(\Rightarrow\left\{{}\begin{matrix}n^3+n⋮d\\n^3+2n⋮d\end{matrix}\right.\)
\(\Rightarrow n⋮d\)
Mà \(n^2+1⋮d\)
\(\Rightarrow\left\{{}\begin{matrix}n^2⋮d\\n^2+1⋮d\end{matrix}\right.\)
\(\Rightarrow1⋮d\)
Vì \(d\in N;1⋮d\Rightarrow d=1\)
\(\RightarrowƯCLN\left(n^3+2n;n^4+3n^2+1\right)\)
Vậy phân số \(\dfrac{n^3+2n}{n^4+3n^2+1}\) tối giản với mọi \(n\in N\)
\(\Rightarrowđpcm\)
~ Chúc bn học tốt ~
