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a, \(\dfrac{x^2}{4}-xy+y^2=\left(\dfrac{x}{2}\right)^2-xy+y^2=\left(\dfrac{x}{2}\right)^2-2.\dfrac{x}{2}.y+y^2\)
\(=\left(\dfrac{x^2}{2}-y\right)^2\)
b, \(x^2+x+\dfrac{1}{4}=x^2+\dfrac{1}{2}.2.x+\left(\dfrac{1}{2}\right)^2=\left(x+\dfrac{1}{2}\right)^2\)
c, \(x^2+2\sqrt{3}x+3=x^2+2\sqrt{3}x+\left(\sqrt{3}\right)^2=\left(x+\sqrt{3}\right)^2\)
d, \(4x^2-1=\left(2x-1\right)\left(2x+1\right)\)
`x^2/4-2*x/2*y+y^2`
`=(x/2-y)^2`
`x^2+x+1/4`
`=x^2+2*x*1/2+(1/2)^2`
`=(x+1/2)^2`
`x^2+2sqrt3x+3`
`=x+2xsqrt3+sqrt3^2`
`=(x+sqrt3)^2`
`4x^2-1`
`=(2x)^2-1`
`=(2x-1)(2x+1)`
Bài giải:
a) 3x - 6y = 3 . x - 3 . 2y = 3(x - 2y)
b) 2525x2 + 5x3 + x2y = x2 (2525 + 5x + y)
c) 14x2y – 21xy2 + 28x2y2 = 7xy . 2x - 7xy . 3y + 7xy . 4xy = 7xy(2x - 3y + 4xy)
d) 2525x(y - 1) - 2525y(y - 1) = 2525(y - 1)(x - y)
e) 10x(x - y) - 8y(y - x) =10x(x - y) - 8y[-(x - y)]
= 10x(x - y) + 8y(x - y)
= 2(x - y)(5x + 4y)
a,\(3x-6y=3\left(x-2y\right)\)
b,\(x^2(\dfrac{2}{5}+5x+y)\)
c,\(7xy\left(2x-3y+4xy\right)\)
d,\(\dfrac{2}{5}x\left(y-1\right)-\dfrac{2}{5}y\left(y-1\right)\)
=\(\dfrac{2}{5}\left(y-1\right)\left(x-y\right)\)
e,\(10x\left(x-y\right)-8y\left(y-x\right)=10x\left(x-y\right)+8y\left(x-y\right)\)
\(2\left(x-y\right)\left(5x+4y\right)\)
\(x^2y^2\left(y-x\right)+y^2z^2\left(z-y\right)-z^2x^2\left(z-x\right)\)
\(=x^2y^2\left(y-x\right)-y^2z^2\left[\left(y-x\right)-\left(z-x\right)\right]-z^2x^2\left(z-x\right)\)
\(=x^2y^2\left(y-x\right)-y^2z^2\left(y-x\right)+y^2z^2\left(z-x\right)-z^2x^2\left(z-x\right)\)
\(=y^2\left(y-x\right)\left(x-z\right)\left(x+z\right)-z^2\left(x-z\right)\left(y-x\right)\left(y+x\right)\)
\(=\left(y-x\right)\left(x-z\right)\left(xy^2+y^2z-z^2y-z^2x\right)\)
Xet \(xy^2+y^2z-z^2y-z^2x=x\left(y-z\right)\left(y+z\right)+yz\left(y-z\right)=\left(y-z\right)\left(xy+yz+zx\right)\)
Vay \(x^2y^2\left(y-x\right)+y^2z^2\left(z-y\right)-z^2x^2\left(z-x\right)=\left(y-x\right)\left(x-z\right)\left(y-z\right)\left(xy+yz+zx\right)\)
\(x^2y^2\left(y-x\right)+y^2z^2\left(z-y\right)-z^2x^2\left(z-x\right)\)
\(=x^2y^3-x^3y^2+y^2z^3-y^3z^2-z^3x^2+z^2x^3\)
\(=y^3\left(x^2-z^2\right)-y^2\left(x^3-z^3\right)+z^2x^2\left(x-z\right)\)
\(=y^3\left(x+z\right)\left(x-z\right)-y^2\left(x-z\right)\left(x^2+xz+z^2\right)+z^2x^2\left(x-z\right)\)
\(=\left(x-z\right)\left(xy^3+y^3z-y^2x^2-y^2xz-y^2z^2+z^2x^2\right)\)
.................
`a, 4x^3 - 16x = 4x(x^2-4) = 4x(x-2)(x+2)`
`b, x^4 - y^4 = (x^2-y^2)(x^2+y^2) = (x-y)(x+y)(x^2+y^2)`
`c, xy^2 + x^2y + 1/4y^3`
`= y(xy + x^2 + 1/4y^2)`
`d, x^2 + 2x - y^2 + 1 = (x+1)^2 - y^2`
`= (x+1+y)(x+1-y)`
1, \(\dfrac{2xy}{x^2-y^2}+\dfrac{x-y}{2x+2y}+\dfrac{y}{y-x}\)
\(=\dfrac{2xy}{x^2-y^2}+\dfrac{\left(x-y\right)\left(y-x\right)+2y\left(x+y\right)}{2\left(x+y\right)\left(y-x\right)}\)
\(=\dfrac{2xy}{x^2-y^2}+\dfrac{xy-x^2-y^2+xy+2xy+2y^2}{2\left(xy-x^2+y^2-xy\right)}\)
\(=\dfrac{2xy}{x^2-y^2}+\dfrac{4xy-x^2+y^2}{2\left(y^2-x^2\right)}\)
\(=\dfrac{2xy}{\left(x-y\right)\left(x+y\right)}+\dfrac{4xy-x^2+y^2}{2\left(y-x\right)\left(x+y\right)}\)
\(=\dfrac{-4xy}{\left(y-x\right)\left(y+x\right)}+\dfrac{4xy-x^2+y^2}{2\left(y-x\right)\left(x+y\right)}\)
\(=\dfrac{y^2-x^2}{2\left(y-x\right)\left(x+y\right)}=\dfrac{\left(y-x\right)\left(x+y\right)}{2\left(y-x\right)\left(x+y\right)}=\dfrac{1}{2}\)
2, \(x^2-y^2-2y-1\)
\(=x^2-\left(y^2+2y+1\right)\)
\(=x^2-\left(y+1\right)^2\)
\(=\left(x-y-1\right)\left(x+y+1\right)\)
\(\dfrac{1}{x-y}-\dfrac{1}{x+y}+\dfrac{2x}{\left(x-y\right)\left(x+y\right)}\\ \dfrac{x+y}{\left(x-y\right)\left(x+y\right)}-\dfrac{x-y}{\left(x-y\right)\left(x+y\right)}+\dfrac{2x}{\left(x-y\right)\left(x+y\right)}\\ \dfrac{x+y-x+y+2x}{\left(x-y\right)\left(x+y\right)}\\ \dfrac{2x+2y}{\left(x-y\right)\left(x+y\right)}\\ \dfrac{2}{x-y}\)
\(\dfrac{1}{x-y}+\dfrac{1}{x+y}+\dfrac{2x}{x^2-y^2}=\dfrac{x+y}{\left(x-y\right)\left(x+y\right)}+\dfrac{x-y}{\left(x-y\right)\left(x+y\right)}+\dfrac{2x}{\left(x-y\right)\left(x+y\right)}=\dfrac{4x}{\left(x-y\right)\left(x+y\right)}\)