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Bài 2:
a: \(3x^2-3xy=3x\left(x-y\right)\)
b: \(x^2-4y^2=\left(x-2y\right)\left(x+2y\right)\)
c: \(3x-3y+xy-y^2=\left(x-y\right)\left(3+y\right)\)
d: \(x^2-y^2+2y-1=\left(x-y+1\right)\left(x+y-1\right)\)
a) \(\dfrac{2x^2-2xy}{x^2+x-xy-y}\) \(\left(x\ne y;x\ne-1\right)\)
\(=\dfrac{2x\left(x-y\right)}{x\left(x+1\right)-y\left(x+1\right)}\)
\(=\dfrac{2x\left(x-y\right)}{\left(x-y\right)\left(x+1\right)}\)
\(=\dfrac{2x}{x+1}\)
b) \(\dfrac{x^2+y^2-z^2+2xy}{x^2-y^2+z^2+2xz}\)
\(=\dfrac{\left(x^2+2xy+y^2\right)-z^2}{\left(x^2+2xz+z^2\right)-y^2}\)
\(=\dfrac{\left(x+y\right)^2-z^2}{\left(x+z\right)^2-y^2}\)
\(=\dfrac{\left(x+y+z\right)\left(x+y-z\right)}{\left(x-y+z\right)\left(x+y+z\right)}\)
\(=\dfrac{x+y-z}{x-y+z}\)
\(P+R=-xy\cdot(x-y)\\\Leftrightarrow R=-xy(x-y)-P\\\Leftrightarrow R=-x^2y+xy^2-(5x^2y-2xy^2+xy-x+y-2)\\\Leftrightarrow R=-x^2y+xy^2-5x^2y+2xy^2-xy+x-y+2\\\Leftrightarrow R=(-x^2y-5x^2y)+(xy^2+2xy^2)-xy+x-y+2\\\Leftrightarrow R=-6x^2y+3xy^2-xy+x-y+2\)
Ta có:
\(P+R=-xy\cdot\left(x-y\right)\)
\(\Leftrightarrow\left(5x^2y-2xy^2+xy-x+y-2\right)+R=-x^2y+xy^2\)
\(\Leftrightarrow R=-x^2y+xy^2-5x^2y+2xy^2+xy+x-y+2\)
\(\Leftrightarrow R=\left(-x^2y-5x^2y\right)+\left(xy^2+2xy^2\right)+xy+x-y+2\)
\(\Leftrightarrow R=-6x^2y+3xy^2+xy+x-y+2\)
`a, (xy^2)/(xy+y) = (xy^2)/(y(x+1))`
`=(xy)/(x+1)`
Vậy `2` cặp phân thức bằng nhau.
`b, (xy-y)/x = (y(x-1))/x = (y^2(x-1))/(xy)`
`(xy-x)/y = (x(y-1))/y = (x^2(y-1))/(xy)`
Vậy `2` đa thức không bằng nhau
\(\frac{x+y}{x-y}.M=\frac{x^2+2xy+y^2}{x^2+xy+y^2}\)
\(\Leftrightarrow M=\frac{x^2+2xy+y^2}{x^2+xy+y^2}.\frac{x+y}{x-y}\)
\(\Leftrightarrow M=\frac{\left(x+y\right)^3}{x^3-y^3}\)
\(\Leftrightarrow M=\frac{x^3+3x^2y+3xy^2+y^3}{x^3-y^3}\)
Sửa:
\(pt\Leftrightarrow M=\frac{x^2+2xy+y^2}{x^2+xy+y^2}.\frac{x-y}{x+y}\)
\(\Leftrightarrow M=\frac{\left(x+y\right)^2.\left(x-y\right)}{\left(x+y\right)\left(x^2+xy+y^2\right)}\)
\(\Leftrightarrow M=\frac{\left(x+y\right)\left(x-y\right)}{\left(x^2+xy+y^2\right)}\)
\(\Leftrightarrow M=\frac{x^2-y^2}{x^2+xy+y^2}\)
b: \(3x+3y-x^2-2xy-y^2\)
\(=3\left(x+y\right)-\left(x+y\right)^2\)
\(=\left(x+y\right)\left(3-x-y\right)\)
Ta có:
\(\dfrac{x-y}{x^3+y^3}\cdot A=\dfrac{x^2-2xy+y^2}{x^2-xy+y^2}\left(x\ne\pm y\right)\)
\(\Leftrightarrow\dfrac{x-y}{\left(x+y\right)\left(x^2-xy+y^2\right)}\cdot A=\dfrac{\left(x-y\right)^2}{x^2-xy+y^2}\)
\(\Leftrightarrow A\cdot\left(x-y\right)=\left(x+y\right)\left(x^2-xy+y^2\right)\cdot\dfrac{\left(x-y\right)^2}{x^2-xy+y^2}\)
\(\Leftrightarrow A\cdot\left(x-y\right)=\left(x+y\right)\left(x-y\right)^2\)
\(\Leftrightarrow A=\dfrac{\left(x+y\right)\left(x-y\right)^2}{x-y}\)
\(\Leftrightarrow A=\left(x+y\right)\left(x-y\right)\)
\(\Leftrightarrow A=x^2-y^2\)