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a) ( xy + 1 )2 - ( x + y )2
= [ ( xy + 1 ) - ( x + y ) ][ ( xy + 1 ) + ( x + y ) ]
= ( xy - x - y + 1 )( xy + x + y + 1 )
b) ( x + y )3 - ( x - y )3
C1. = x3 + 3x2y + 3xy2 + y3 - ( x3 - 3x2y + 3xy2 - y3 )
= x3 + 3x2y + 3xy2 + y3 - x3 + 3x2y - 3xy2 + y3
= 6x2y + 2y3
= 2y( 3x2 + y2 )
C2. = [ ( x + y ) - ( x - y ) ][ ( x + y )2 + ( x + y )( x - y ) + ( x - y )2 ]
= ( x + y - x + y )( x2 + 2xy + y2 + x2 - y2 + x2 - 2xy + y2 )
= 2y( 3x2 + y2 )
c) 3x4y2 + 3x3y2 + 3xy2 + 3y2
= 3( x4y2 + x3y2 + xy2 + y2 )
= 3[ ( x4y2 + x3y2 ) + ( xy2 + y2 ) ]
= 3[ x3y2( x + 1 ) + y2( x + 1 ) ]
= 3( x + 1 )( x3y2 + y2 )
= 3y2( x + 1 )( x3 + 1 )
= 3y2( x + 1 )( x + 1 )( x2 - x + 1 )
= 3y2( x + 1 )2( x2 - x + 1 )
d) 4( x2 - y2 ) - 8( x - ay ) - 4( a2 - 1 )
= 4[ ( x2 - y2 ) - 2( x - ay ) - ( a2 - 1 )
= 4( x2 - y2 - 2x + 2ay - a2 + 1 )
= 4[ ( x2 - 2x + 1 ) - ( y2 - 2ay + a2 ) ]
= 4[ ( x - 1 )2 - ( y - a )2 ]
= 4[ ( x - 1 ) - ( y - a ) ][ ( x - 1 ) + ( y + a ) ]
= 4( x - y + a - 1 )( x + y + a - 1 )
a) 36 - 4a2 + 20ab - 25b2 = 36 - ( 4a2 - 20ab + 25b2 ) = 62 - ( 2a - 5b )2 = ( 6 - 2a + 5b )( 6 + 2a - 5b )
b) ( xy + 4 )2 - 4( x + y )2 = ( xy + 4 )2 - 22( x + y )2 = ( xy + 4 )2 - [ 2( x + y ) ]2
= ( xy + 4 )2 - ( 2x + 2y )2 = ( xy + 4 - 2x - 2y )( xy + 4 + 2x + 2y )
= [ x( y - 2 ) - 2( y - 2 ) ][ x( y + 2 ) + 2( y + 2 ) ]
= ( y - 2 )( x - 2 )( y + 2 )( x + 2 )
c) x2 + y2 - x2y2 + xy - x - y
= ( x2 - x2y2 ) + ( y2 - y ) + ( xy - x )
= x2( 1 - y2 ) + y( y - 1 ) + x( y - 1 )
= x2( 1 - y )( 1 + y ) - y( 1 - y ) - x( 1 - y )
= ( 1 - y )[ x2( 1 + y ) - y - x ) ]
= ( 1 - y )( x2 + x2y - y - x )
= ( 1 - y )[ ( x2 - x ) + ( x2y - y ) ]
= ( 1 - y )[ x( x - 1 ) + y( x2 - 1 ) ]
= ( 1 - y )[ x( x - 1 ) + y( x - 1 )( x + 1 ) ]
= ( 1 - y )( x - 1 )[ x + y( x + 1 ) ]
= ( 1 - y )( x - 1 )( x + xy + y )
d) 3x + 3y - x2 - 2xy - y2
= 3( x + y ) - ( x2 + 2xy + y2 )
= 3( x + y ) - ( x + y )2
= ( x + y )( 3 - x - y )
e) ( 2xy + 1 )2 - ( 2x + y )2
= ( 2xy + 1 - 2x - y )( 2xy + 1 + 2x + y )
= [ ( 2xy - 2x ) - ( y - 1 ) ][ ( 2xy + 2x ) + ( y + 1 ) ]
= [ 2x( y - 1 ) - ( y - 1 ) ][ 2x( y + 1 ) + ( y + 1 ) ]
= ( y - 1 )( 2x - 1 )( y + 1 )( 2x + 1 )
a) \(36-4a^2+20ab-25b^2\)
\(=36-\left(4a^2-20ab+25b^2\right)\)
\(=36-\left(2a-5b\right)^2\)
\(=\left(6-2a+5b\right)\left(6+2a-5b\right)\)
b) \(\left(xy+4\right)^2-4\left(x+y\right)^2\)
\(=\left(xy+4-2x-2y\right)\left(xy+4+2x+2y\right)\)
\(=\left[x\left(y-2\right)-2\left(y-2\right)\right]\left[x\left(y+2\right)+2\left(y+2\right)\right]\)
\(=\left(x+2\right)\left(x-2\right)\left(y+2\right)\left(y-2\right)\)
c) \(x^2+y^2-x^2y^2+xy-x-y\)
\(=-\left(x^2y^2-x^2\right)+\left(y^2-y\right)+\left(xy-x\right)\)
\(=-x^2\left(y-1\right)\left(y+1\right)+y\left(y-1\right)+x\left(y-1\right)\)
\(=\left(y-1\right)\left(-x^2y-x^2+y+x\right)\)
\(=\left(1-y\right)\left[\left(x^2y-y\right)+\left(x^2-x\right)\right]\)
\(=\left(1-y\right)\left(x-1\right)\left(xy+y+x\right)\)
\(a,xy+1-x-y\)
\(=\left(xy-y\right)+\left(1-x\right)\)
\(=y\left(x-1\right)- \left(x-1\right)\)
\(=\left(x-1\right)\left(y-1\right)\)
\(b,ax+ay-3x-3y\)
\(=a\left(x+y\right)-3\left(x+y\right)\)
\(=\left(x+y\right)\left(a-3\right)\)
\(c,x^3-2x^2+2x-4\)
\(=x^2\left(x-2\right)+2\left(x-2\right)\)
\(=\left(x^2+2\right)\left(x-2\right)\)
\(d,x^2+ab+ax+bx\)
\(=\left(x^2+ax\right)+\left(ab+bx\right)\)
\(=x\left(a+x\right)+b\left(a+x\right)\)
\(=\left(a+x\right)\left(b+x\right)\)
\(e,16-x^2+2xy-y^2\)
\(=4^2-\left(x^2-2xy+y^2\right)\)
\(=4^2-\left(x-y\right)^2\)
\(=\left(4-x+y\right)\left(4+x-y\right)\)
\(3x\left(x-5\right)-x\left(4+3x\right)=43\)
\(\Leftrightarrow3x^2-15x-4x-3x^2=43\)
\(\Leftrightarrow-19x=43\)
\(\Leftrightarrow x=\frac{-43}{19}\)
a) \(\left(xy+1\right)^2-\left(x+y\right)\)
\(=\left(xy+1-x-y\right)\left(xy+1+x+y\right)\)
\(=\left[x\left(y-1\right)-\left(y-1\right)\right]\left[x\left(y+1\right)+\left(y+1\right)\right]\)
\(=\left(x-1\right)\left(y-1\right)\left(x+1\right)\left(y+1\right)\)
b) \(\left(x+y\right)^3-\left(x-y\right)^3\)
\(=\left(x+y-x+y\right)\left[\left(x+y\right)^2+\left(x-y\right)\left(x+y\right)+\left(x-y\right)^2\right]\)
\(=2y\left(x^2+2xy+y^2+x^2-y^2+x^2-2xy+y^2\right)\)
\(=2y\left(3x^2+y^2\right)\)