Question

[Felix MC-Gamer]

Giải giùm với mấy chế ơi

1: Hỏi: \(n\in Z\) thì \(n^3-7n+2018\) có chia hết cho2018 không?

2: \(n\in N\) chứng mình các phân số sau tối giản:

a) \(\dfrac{4n+1}{5n+1}\); b) \(\dfrac{12n+1}{30n+1}\)

3: rút gọn: \(C=\dfrac{x-x^3}{x^2+1}\left(\dfrac{1}{1+2x+x^2}+\dfrac{1}{1-x^2}\right)+\dfrac{1}{1+x}\)

4:chứng minh: \(\left(\dfrac{x+2}{x+1}-\dfrac{4\left(y+1\right)}{y+2}\right):\left(\dfrac{x^2\left(y+1\right)}{x+1}-\dfrac{y^2\left(x+2\right)}{y+2}\right)=\dfrac{1}{y-x}\)

Answer 1

[Duy Đỗ Ngọc Tuấn]

C=\(\dfrac{x-x^3}{x^2+1}\left(\dfrac{1}{1+2x+x^2}+\dfrac{1}{1-x^2}\right)+\dfrac{1}{1+x}\)

\(=\dfrac{x\left(1-x^2\right)}{x^2+1}\left(\dfrac{1}{\left(1+x\right)^2}+\dfrac{1}{\left(1-x\right)\left(1+x\right)}\right)+\dfrac{1}{1+x}\)

\(=\dfrac{x\left(1-x\right)\left(1+x\right)}{x^2+1}\left(\dfrac{1-x+1+x}{\left(1-x\right)\left(1+x\right)^2}\right)+\dfrac{1}{1+x}\)

\(=\dfrac{x\left(1-x\right)\left(1+x\right).2}{\left(x^2+1\right)\left(1-x\right)\left(1+x^2\right)}+\dfrac{1}{1+x}\)

\(=\dfrac{2x}{\left(x^2+1\right)\left(1+x\right)}+\dfrac{1}{1+x}\)

\(=\dfrac{2x+\left(x^2+1\right)}{\left(x^2+1\right)\left(1+x\right)}\)

\(=\dfrac{2x+x^2+1}{\left(x^2+1\right)\left(x+1\right)}\)

\(=\dfrac{x^2+2x+1}{\left(x^2+1\right)\left(x+1\right)}\)

\(=\dfrac{\left(x+1\right)^2}{\left(x^2+1\right)\left(x +1\right)}\)

\(=\dfrac{x+1}{x^2+1}\)