Question

\(\frac{1}{\left(x+1\right)\left(x+2\right)}-\frac{2}{\left(x+2\right)^2}+\frac{1}{\left(x+2\right)\left(x+3\right)}\)rút gọn biểu thức trên

Answer 1

\(\frac{1}{\left(x+1\right)\left(x+2\right)}-\frac{2}{\left(x+2\right)^2}+\frac{1}{\left(x+2\right)\left(x+3\right)}\)

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

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

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

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

Answer 2

ĐKXD: x\(\ne\)-1,-2,-3

Ta có

\(\frac{1}{\left(x+1\right)\left(x+2\right)}\)-\(\frac{2}{\left(x+2\right)^2}\)+\(\frac{1}{\left(x+2\right)\left(x+3\right)}\)

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

=\(\frac{\left(x+2\right)\left(x+3+x+1\right)-2\left(x^2+4x+3\right)}{\left(x+1\right)\left(x+2\right)^2\left(x+3\right)}\)

=\(\frac{\left(x+2\right)\left(2x+4\right)-2x^2-8x-6}{\left(x+1\right)\left(x+2\right)^2\left(x+3\right)}\)

=\(\frac{2x^2+8x+8-2x^2-8x-6}{\left(x+1\right)\left(x+2\right)^2\left(x+3\right)}\)

=\(\frac{2}{\left(x+1\right)\left(x+2\right)^2\left(x+3\right)}\)

Chúc bạn học tốt

Answer 3

\(ĐKXĐ:\hept{\begin{cases}x\ne-1\\x\ne-2\\x\ne-3\end{cases}}\)

\(\frac{1}{\left(x+1\right)\left(x+2\right)}-\frac{2}{\left(x+2\right)^2}+\frac{1}{\left(x+2\right)\left(x+3\right)}\)

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

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

\(=\frac{x^2+5x+6-2\left(x^2+4x+3\right)+x^2+3x+2}{\left(x+1\right)\left(x+2\right)^2\left(x+3\right)}\)\(=\frac{x^2+5x+6-2x^2-8x-6+x^2+3x+2}{\left(x+1\right)\left(x+2\right)^2\left(x+3\right)}\)

\(=\frac{2}{\left(x+1\right)\left(x+2\right)^2\left(x+3\right)}\)