Question

[Nguyễn Đức Anh]

Cho biểu thức 

P=\(\frac{2x+3}{x+1}\)-\(\frac{x+2}{x-1}\)+\(\frac{3x+5}{x^2-1}\)

a, Tìm ĐKXĐ và rút gọn P

b,Tìm x để P thuộc Z

Answer 1

[Minh Nguyen]

\(ĐKXĐ:x\ne\pm1\)

a) \(P=\frac{2x+3}{x+1}-\frac{x+2}{x-1}+\frac{3x+5}{x^2-1}\)

\(\Leftrightarrow P=\frac{\left(2x+3\right)\left(x-1\right)-\left(x+2\right)\left(x+1\right)+3x+5}{\left(x-1\right)\left(x+1\right)}\)

\(\Leftrightarrow P=\frac{2x^2+x-3-x^2-3x-2+3x+5}{\left(x-1\right)\left(x+1\right)}\)

\(\Leftrightarrow P=\frac{x^2+x}{\left(x-1\right)\left(x+1\right)}\)

\(\Leftrightarrow P=\frac{x\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}\)

\(\Leftrightarrow P=\frac{x}{x-1}\)

b) Để \(P\inℤ\)

\(\Leftrightarrow x⋮x-1\)

\(\Leftrightarrow x-1+1⋮x-1\)

\(\Leftrightarrow1⋮x-1\)

\(\Leftrightarrow x-1\inƯ\left(1\right)=\left\{\pm1\right\}\)

\(\Leftrightarrow x\in\left\{0;2\right\}\)

Vậy để \(A\inℤ\Leftrightarrow x\in\left\{0;2\right\}\)