Question

giúp em giải bài này với ạ

Bài 3. Thực hiện phép trừ các phân thức sau với điều kiện các phân thức đều xác định

a) \(\frac{3x}{x+1} - \frac{1}{1+x}\)

b) \(\frac{3y-2x}{x-2y} - \frac{x-y}{2y-x}\)

c) \(\frac{1}{x-2} - \frac{1}{x+1}\)

d) \(\frac{12}{x^2-9} - \frac{2}{x-3}\)

e) \(\frac{1}{xy-x^2} - \frac{1}{y^2-xy}\)

f) \(\frac{x}{x^2-25} - \frac{x-5}{x^2+5x}\)

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

h) \(\frac{x}{x-1} - \frac{2}{x+1} - \frac{2x}{x^2-1}\)

i) \(\frac{8x^2}{4x^3-8x^2} - \frac{3x}{3x^2-12} - \frac{1}{x+2}\)

j) \(\frac{x^3+1}{x^2-9} - \frac{3}{x+3}\)

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

l) \(\frac{x+1}{x+1} - 1\)

Answer 1

a: \(\frac{3x}{x+1}-\frac{1}{1+x}=\frac{3x}{x+1}-\frac{1}{x+1}=\frac{3x-1}{x+1}\)

b: \(\frac{3y-2x}{x-2y}-\frac{x-y}{2y-x}\)

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

=-1

c: \(\frac{1}{x-2}-\frac{1}{x+1}\)

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

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

d: \(\frac{12}{x^2-9}-\frac{2}{x-3}\)

\(=\frac{12}{\left(x-3\right)\left(x+3\right)}-\frac{2}{x-3}\)

\(=\frac{12-2\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}=\frac{12-2x-6}{\left(x-3\right)\left(x+3\right)}\)

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

e: \(\frac{1}{xy-x^2}-\frac{1}{y^2-xy}\)

\(=\frac{1}{x\left(y-x\right)}-\frac{1}{y\left(y-x\right)}\)

\(=\frac{y-x}{xy\left(y-x\right)}=\frac{1}{xy}\)

f: \(\frac{x}{x^2-25}-\frac{x-5}{x^2+5x}\)

\(=\frac{x}{\left(x-5\right)\left(x+5\right)}-\frac{x-5}{x\left(x+5\right)}\)

\(=\frac{x^2-\left(x-5\right)}{x\left(x+5\right)\left(x-5\right)}=\frac{\left(x-x+5\right)\left(x+x-5\right)}{x\left(x+5\right)\left(x-5\right)}=\frac{5\left(2x-5\right)}{x\left(x+5\right)\left(x-5\right)}\)

g: \(\frac{2x^3+x^2-x}{x^3-1}-\frac{2x-1}{x-1}\)

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

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

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

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

h: \(\frac{x}{x-1}-\frac{2}{x+1}-\frac{2}{x^2-1}\)

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

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

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

i: \(\frac{8x^2}{4x^3-8x^2}-\frac{3x}{3x^2-12}-\frac{1}{x+2}\)

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

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

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

j: \(\frac{x^3+1}{x^2-9}-\frac{3}{x+3}-x\)

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

\(=\frac{x^3+1-3\left(x-3\right)-x\left(x^2-9\right)}{\left(x-3\right)\left(x+3\right)}=\frac{x^3+1-3x+9-x^3+9x}{\left(x-3\right)\left(x+3\right)}=\frac{6x+10}{\left(x-3\right)\cdot\left(x+3\right)}\)

k: \(\frac{2x}{x^2-1}-\frac{3}{2x+2}+\frac{1}{2-2x}\)

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

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

l: \(x+\frac{1}{x+1}-1=x-1+\frac{1}{x+1}=\frac{\left(x-1\right)\left(x+1\right)+1}{x+1}\)

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