\(3x^{n-2}\left(x^{n+2}-y^{n+2}\right)+y^{n+2}\left(3x^{n-2}-y^{2-2}\right)\)

\(=3x^{2n}-3x^{n-2}y^{n+2}+y^{n+2}\left(3x^{n-2}-y^{n-2}\right)\)

\(=3x^{2n}-3x^{n-2}y^{n+2}+3x^{n-2}y^{n+2}-y^{2n}\)

\(=3x^{2n}-y^{2n}\)

P/s: Mk ko rõ đề nên làm vậy nhé!

Đề bài chắc là đơn giản tỉ lệ thức(rút gọn) nên mình làm luôn nha:

\(3x^{n-2}\left(x^{n+2}-y^{n+2}\right)+y^{n+2}\left(3x^{n-2}-y^{n-2}\right)\)

\(=3x^{2n}-3xy^{2n}+3xy^2-y^{2n}\)

\(=3x^{2n}-y^{2n}\)

3x²ⁿ - y²ⁿ

Câu 1: 

\(\dfrac{A}{B}=\dfrac{4x^{n+1}y^2}{3x^3y^{n-1}}=\dfrac{4}{3}x^{n-2}y^{2-n+1}=\dfrac{4}{3}x^{n-2}y^{3-n}\)

Để A chia hết cho B thì \(\left\{{}\begin{matrix}n-2>=0\\3-n>=0\end{matrix}\right.\Leftrightarrow2\le n\le3\)

Bài 2: 

\(=\dfrac{\left(x+y\right)\left(x^2-xy+y^2\right)-2\left(x+y\right)\left(x-y\right)+3\left(x+y\right)^2}{x+y}\)

\(=x^2-xy+y^2-2\left(x-y\right)+3\left(x+y\right)\)

\(=x^2-xy+y^2-2x+2y+3x+3y\)

\(=x^2-xy+y^2+x+5y\)

\(=3^{2n}-3x^{n-2}y^{n+2}+3x^{n-2}y^{n-2}-y^{2n}\)

\(=3x^{2n}-y^{2n}\)

Lời giải:

Ta có:

\(A=3x^{n-2}(x^{n+2}-y^{n+2})+y^{n+2}(3x^{n-2}-y^{n-2})\)

\(=3x^{n-2}.x^{n+2}-3x^{n-2}y^{n+2}+3y^{n+2}x^{n-2}-y^{n+2}.y^{n-2}\)

\(=3x^{n-2+n+2}-y^{n+2+n-2}=3x^{2n}-y^{2n}\)

a) \(\left(3x^{n+1}-y^{n-1}\right)-3\left(x^{n+1}+5y^{n-1}\right)-4\left(x^{n+1}+2y^{n-1}\right)\)

\(=3x^{n+1}-y^{n-1}-3x^{n+1}-15y^{n-1}+4x^{n+1}+8y^{n-1}\)

\(=-8y^{n-1}+4x^{n+1}\)

b) \(\left(\dfrac{3}{4}x^{n+1}-\dfrac{1}{2}y^n\right)\cdot2xy-\left(\dfrac{2}{3}x^{n+1}-\dfrac{5}{6}y^n\right)\cdot7xy\)

\(=\dfrac{3}{2}x^{n+2}y-xy^{n+1}+\left(-\dfrac{2}{3}x^{n+1}-\dfrac{5}{6}y^n\right)\cdot7xy\)

\(=\dfrac{3}{2}x^{n+2}y-xy^{n+1}-\dfrac{14}{3}x^{n+2}y+\dfrac{35}{6}xy^{n+1}\)

\(=-\dfrac{19}{6}x^{n+2}y+\dfrac{29}{6}xy^{n+1}\)

a)\(\left(3x^{n+1}-y^{n-1}\right)-3\left(x^{n+1}+5y^{n-1}\right)+4\left(x^{n+1}+2y^{n-1}\right)\)

\(=3x^{n+1}-y^{n-1}-3x^{n+1}-15y^{n-1}+4x^{n+1}+8y^{n-1}\)

\(=4x^{n+1}-8y^{n-1}\) \(\left(=4\left(x^{n+1}-2y^{n-1}\right)\right)\)