Bài 4: Quy đồng mẫu thức nhiều phân thức

Bài 3: 

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

\(\dfrac{2}{2x-4}=\dfrac{1}{x-2}=\dfrac{x\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)}\)

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

Bài 5:

\(a,\dfrac{2}{2x-4}=\dfrac{2}{2\left(x-2\right)}=\dfrac{1}{x-2};\dfrac{3}{3x-6}=\dfrac{3}{3\left(x-2\right)}=\dfrac{1}{x-2}\\ b,\dfrac{1}{x+4}=\dfrac{2\left(x-4\right)}{2\left(x+4\right)\left(x-4\right)};\dfrac{1}{2x+8}=\dfrac{x-4}{2\left(x+4\right)\left(x-4\right)}\\ \dfrac{3}{x-4}=\dfrac{6\left(x+4\right)}{2\left(x-4\right)\left(x+4\right)}\\ c,\dfrac{1}{x^2-1}=\dfrac{1}{\left(x-1\right)\left(x+1\right)};\dfrac{2}{x-1}=\dfrac{2\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}\\ \dfrac{2}{x+1}=\dfrac{2\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}\\ d,\dfrac{1}{2x}=\dfrac{x-2}{2x\left(x-2\right)};\dfrac{2}{x-2}=\dfrac{4x}{2x\left(x-2\right)};\dfrac{3}{2x\left(x-2\right)}\text{ giữ nguyên}\)

Bài 4:

\(a,\dfrac{x^2-4x+4}{x^2-2x}=\dfrac{\left(x-2\right)^2}{x\left(x-2\right)}=\dfrac{x-2}{x}=\dfrac{\left(x-2\right)\left(x-1\right)}{x\left(x-1\right)}\\ \dfrac{x+1}{x^2-1}=\dfrac{1}{x-1}=\dfrac{x}{x\left(x-1\right)}\\ b,\dfrac{x^3-2^3}{x^2-4}=\dfrac{\left(x-2\right)\left(x^2+2x+4\right)}{\left(x-2\right)\left(x+2\right)}=\dfrac{x^2+2x+4}{x+2};\dfrac{3}{x+2}\text{ giữ nguyên}\)

i: \(=\dfrac{1}{a}-\dfrac{1}{a+1}+\dfrac{1}{a+1}-\dfrac{1}{a+2}+\dfrac{1}{a+2}-\dfrac{1}{a+3}+\dfrac{1}{a+3}=\dfrac{1}{a}\)

\(=\dfrac{x+1}{2\left(x+3\right)}+\dfrac{2x+3}{x\left(x+3\right)}=\dfrac{x\left(x+1\right)+\left[2\left(2x+3\right)\right]}{2x\left(x+3\right)}=\dfrac{x^2+x+4x+6}{2x\left(x+3\right)}=\dfrac{x^2+5x+6}{2x\left(x+3\right)}=\dfrac{\left(x+2\right)\left(x+3\right)}{2x\left(x+3\right)}=\dfrac{x+2}{2x}\)

\(\dfrac{x+1}{2x+6}\) + \(\dfrac{2x+3}{x^2+3x}\) = \(\dfrac{x+1}{2\left(x+3\right)}\) + \(\dfrac{2x+3}{x\left(x+3\right)}\)

MTC: 2x ( x + 3 )  

=> \(\dfrac{x\left(x+1\right)}{2x\left(x+3\right)}\) + \(\dfrac{2\left(2x+3\right)}{2x\left(x+3\right)}\) = \(\dfrac{x^2+x+4x+6}{2x\left(x+3\right)}\) = \(\dfrac{x^2+5x+6}{2x\left(x+3\right)}\) = \(\dfrac{\left(x+2\right)\left(x+3\right)}{2x\left(x+3\right)}\) = \(\dfrac{x+2}{2x}\)

d: \(\Leftrightarrow\dfrac{\left(x+2\right)^2}{\left(x+2\right)\left(x-2\right)}=\dfrac{\left(x+1\right)\left(x+2\right)}{A}\)

hay A=x-2

\(\left\{{}\begin{matrix}\dfrac{x}{x^3+1}=\dfrac{x}{\left(x+1\right)\left(x^2-x+1\right)}=\dfrac{x^2}{x\left(x+1\right)\left(x^2-x+1\right)}\\\dfrac{x+1}{x^2+x}=\dfrac{x+1}{x\left(x+1\right)}=\dfrac{1}{x}=\dfrac{x^3+1}{x\left(x+1\right)\left(x^2-x+1\right)}\\\dfrac{x+2}{x^2-x+1}=\dfrac{x\left(x+1\right)\left(x+2\right)}{x\left(x+1\right)\left(x^2-x+1\right)}=\dfrac{x^3+3x^2+2x}{x\left(x+1\right)\left(x^2-x+1\right)}\end{matrix}\right.\)

\(a,\dfrac{13z}{63x^2y^3}=\dfrac{65z^3}{315x^2y^3z^2};\dfrac{-y}{15xz^2}=\dfrac{-21xy^4}{315x^2y^3z^2};\dfrac{2x}{9y^2z}=\dfrac{70x^3yz}{315x^2y^3z}\\ b,\dfrac{x}{x-y}=\dfrac{-x\left(y-x\right)^2}{\left(y-x\right)^3};\dfrac{y}{\left(x-y\right)^2}=\dfrac{y}{\left(y-x\right)^2}=\dfrac{y}{\left(y-x\right)^3}\\ c,\dfrac{1}{2x+4}=\dfrac{x-2}{2\left(x+2\right)\left(x-2\right)};\dfrac{x}{2x-4}=\dfrac{x\left(x+2\right)}{2\left(x-2\right)\left(x+2\right)};\dfrac{3}{4-x^2}=-\dfrac{6}{2\left(x-2\right)\left(x+2\right)}\)

\(a,\dfrac{1}{x+2}=\dfrac{x\left(2-x\right)}{x\left(2-x\right)\left(2+x\right)}\\ \dfrac{8}{2x-x^2}=\dfrac{8}{x\left(2-x\right)}=\dfrac{8\left(2+x\right)}{x\left(2-x\right)\left(2+x\right)}\)

\(b,\dfrac{2-x}{x^2-9}=\dfrac{2-x}{\left(x-3\right)\left(x+3\right)}=\dfrac{x\left(2-x\right)}{x\left(x-3\right)\left(x+3\right)}\\ \dfrac{-1}{x^2+3x}=\dfrac{-1}{x\left(x+3\right)}=\dfrac{-\left(x-3\right)}{x\left(x-3\right)\left(x+3\right)}=\dfrac{3-x}{x\left(x-3\right)\left(x+3\right)}\)