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\(x^4-y^4+2x^3y-2xy^3\)
\(=\left(x^2+y^2\right)\left(x^2-y^2\right)+2xy\left(x^2-y^2\right)\)
\(=\left(x^2-y^2\right)\left(x^2+y^2+2xy\right)\)
\(=\left(x-y\right)\left(x+y\right)\left(x+y\right)^2\)
\(=\left(x-y\right)\left(x+y\right)^3\)
\(x^4-y^4+2x^3y-2xy^3\\ =\left(x^2\right)^2-\left(y^2\right)^2+2xy\left(x^2-y^2\right)\\ =\left(x^2-y^2\right)\left(x^2+y^2\right)+2xy\left(x^2-y^2\right)\\ =\left(x^2-y^2\right)\left(x^2+y^2+2xy\right)\\ =\left(x-y\right)\left(x+y\right)\left(x+y\right)^2\\ =\left(x-y\right)\left(x+y\right)^3\)
`a)x^3-8x^2+16x`
`=x(x^2-8x+16)`
`=x(x-4)^2`
`b)x^2+4y^2+2x-4y-4xy-24`
`=(x-2y)^2+2(x-2y)-24`
`=(x-2y)^2-4(x-2y)+6(x-2y)-24`
`=(x-2y-4)(x-2y+6)`
`c)x^4+x^3-x^2-2x-2`
`=x^4-2x^2+x^3-2x+x^2-2`
`=x^2(x^2-2)+x(x^2-2)+x^2-2`
`=(x^2-2)(x^2+x+1)`
a: \(x^2+4x+4=x^2+2\cdot x\cdot2+2^2=\left(x+2\right)^2\)
b: \(4x^2-4x+1=\left(2x\right)^2-2\cdot2x\cdot1+1^2=\left(2x-1\right)^2\)
c: \(2x-1-x^2\)
\(=-\left(x^2-2x+1\right)=-\left(x-1\right)^2\)
d: \(x^2+x+\dfrac{1}{4}=x^2+2\cdot x\cdot\dfrac{1}{2}+\left(\dfrac{1}{2}\right)^2=\left(x+\dfrac{1}{2}\right)^2\)
e: \(9-x^2=3^2-x^2=\left(3-x\right)\left(3+x\right)\)
g: \(\left(x+5\right)^2-4x^2=\left(x+5+2x\right)\left(x+5-2x\right)\)
\(=\left(5-x\right)\left(5+3x\right)\)
h: \(\left(x+1\right)^2-\left(2x-1\right)^2\)
\(=\left(x+1+2x-1\right)\left(x+1-2x+1\right)\)
\(=3x\left(-x+2\right)\)
i: \(=x^2y^2-4xy+4-3\)
\(=\left(xy-2\right)^2-3=\left(xy-2-\sqrt{3}\right)\left(xy-2+\sqrt{3}\right)\)
k: \(=y^2-\left(x-1\right)^2\)
\(=\left(y-x+1\right)\left(y+x-1\right)\)
l: \(=x^3+3\cdot x^2\cdot2+3\cdot x\cdot2^2+2^3=\left(x+2\right)^3\)
m: \(=\left(2x\right)^3-3\cdot\left(2x\right)^2\cdot y+3\cdot2x\cdot y^2-y^3=\left(2x-y\right)^3\)
a) (x - y)(x + y + 3). b) (x + y - 2xy)(2 + y + 2xy).
c) x 2 (x + l)( x 3 - x 2 + 2). d) (x – 1 - y)[ ( x - 1 ) 2 + ( x - 1 ) y + y 2 ].
\(a,=\left(3x+1\right)^2-y^2=\left(3x-y+1\right)\left(3x+y+1\right)\\ b,=x\left(x^2-5x+6\right)=x\left(x^2-2x-3x+6\right)=x\left(x-2\right)\left(x-3\right)\)
\(=2xy\left(x^2-y^2-2y-1\right)\)
\(=2xy\left[x^2-\left(y^2+2y+1\right)\right]\)
\(=2xy\left[x^2-\left(y+1\right)^2\right]\)
\(=2xy\left(x-y-1\right)\left(x+y+1\right)\)
\(2x^3y-2xy^3-4xy^2-2xy\)
\(=2xy\left(x^2-y^2-2y-1\right)\)
\(=2xy\left[x^2-\left(y+1\right)^2\right]\)
\(=2xy\left(x-y-1\right)\left(x+y+1\right)\)
2x3y – 2xy3 – 4xy2 – 2xy
= 2xy(x2 - y2 - 2y - 1)
= 2xy[x2 - (y2 + 2y + 1)]
= 2xy[x2 - (y + 1)2 ]
= 2xy(x + y + 1)(x - y - 1)
2) 9x2+ 12x+ 4
<=>(3x)2+ 2.3x.2+ 22 <=>(3x+ 2)2
3) 4x4+ 20x2+ 25
<=>(2x2)2+ 2.2x2.5+ 52 <=>(2x2+5)2
4) 25x2- 20xy+ 4y2
<=> (5x)2- 2.5x.2y+ (2y)2<=> (5x-2y)2
5) 9x4- 12x2y+ 4y2
<=> (3x2)2- 2.3x2.2.y+ (2y)2<=> (3x2- 2y)2
6) 4x4- 16x2y3+ 16y6
<=> (2x2)2- 2.2x2.4y3+ (4y3)2<=> (2x2- 4y3)2
7) 9x4- 12x5+ 4x6
<=> (3x2)2- 2.3x2.2x3+ (2x3)2<=> (3x2- 2x3)2