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a) ( 5x - y )( 25x2 + 5xy + y2 ) = ( 5x )3 - y3 = 125x3 - y3
b) ( x - 3 )( x2 + 3x + 9 ) - ( 54 + x3 ) = x3 - 33 - 54 - x3 = -27 - 54 = -81
c) ( 2x + y )( 4x2 - 2xy + y2 ) - ( 2x - y )( 4x2 + 2xy + y2 ) = ( 2x )3 + y3 - [ ( 2x )3 - y3 ]= 8x3 + y3 - 8x3 + y3 = 2y3
d) ( x + y )2 + ( x - y )2 + ( x + y )( x - y ) - 3x2 = x2 + 2xy + y2 + x2 - 2xy + y2 + x2 - y2 - 3x2 = y2
e) ( x - 3 )3 - ( x - 3 )( x2 + 3x + 9 ) + 6( x + 1 )2
= x3 - 9x2 + 27x - 27 - ( x3 - 33 ) + 6( x2 + 2x + 1 )
= x3 - 9x2 + 27x - 27 - x3 + 27 + 6x2 + 12x + 6
= -3x2 + 39x + 6
= -3( x2 - 13x - 2 )
f) ( x + y )( x2 - xy + y2 ) + ( x - y )( x2 + xy + y2 ) - 2x3
= x3 + y3 + x3 - y3 - 2x3
= 0
g) x2 + 2x( y + 1 ) + y2 + 2y + 1
= x2 + 2x( y + 1 ) + ( y2 + 2y + 1 )
= x2 + 2x( y + 1 ) + ( y + 1 )2
= ( x + y + 1 )2
= [ ( x + y ) + 1 ]2
= ( x + y )2 + 2( x + y ) + 1
= x2 + 2xy + y2 + 2x + 2y + 1
\(1,\)
\(\left(x^2-9y^2\right)\left(4x+12y\right)\)
\(=\left(x-3y\right)\left(x+3y\right)-4\left(x+3y\right)\)
\(=\left(x+3y\right)\left(x-3y-4\right)\)
\(3,\)
\(-x^2+2xy-y^2+25\)
\(=-\left(x^2-2xy+y^2\right)+25\)
\(=25-\left(x-y\right)^2\)
\(=5^2-\left(x-y\right)^2\)
\(=\left(5-x+y\right)\left(5+x-y\right)\)
h, \(27x^3-8=\left(3x-2\right)\left(9x^2+6x+4\right)\)
\(\Rightarrow\left(27x^3-8\right):\left(3x-2\right)\\ =\left(3x-2\right)\left(9x^2+6x+4\right):\left(3x-2\right)\\ =9x^2+6x+4\)
g, \(x^4-2x^2+1=\left(x^2-1\right)^2\)
\(\Rightarrow\left(x^4-2x^2+1\right):\left(1-x^2\right)\\ =\left(x^2-1\right)^2:\left(1-x^2\right)\\ =x^2-1\)
a)<=>
A,=(x+y)(x-y)=x^2-y^2
x=(-1/2)^5:(1/2)^4=-1/2
x^2=1/4
y=8^2/(-2)^5=-2
y^2=4
A=1/4-4=-15/4
1) x2 + 2xy + y2 + 2x + 2y - 15
= ( x2 + 2xy + y2 + 2x + 2y + 1 ) - 16
= [ ( x2 + 2xy + y2 ) + 2( x + y ) + 12 ] - 42
= [ ( x + y )2 + 2( x + y ) + 12 ] - 42
= ( x + y + 1 )2 - 42
= ( x + y + 1 - 4 )( x + y + 1 + 4 )
= ( x + y - 3 )( x + y + 5 )
2) x4 - x3 + x2 - 1
= ( x4 - x3 ) + ( x2 - 1 )
= x3( x - 1 ) + ( x - 1 )( x + 1 )
= ( x - 1 )[ x3 + ( x + 1 ) ]
= ( x - 1 )( x3 + x + 1 )
1) x2 + x2y - y - 1
= x2( 1 + y ) - ( 1 + y )
= ( 1 + y )( x2 - 1 )
= ( 1 + y )( x - 1 )( x + 1 )
2) x2 + y2 - 2xy - 25
= ( x2 - 2xy + y2 ) - 25
= ( x - y )2 - 52
= ( x - y - 5 )( x - y + 5 )
3) ( 2x - 1 )( x2 + 2x - 1 ) - ( 1 - 2x )( x - 3 )
= ( 2x - 1 )( x2 + 2x - 1 ) + ( 2x - 1 )( x - 3 )
= ( 2x - 1 )( x2 + 2x - 1 + x - 3 )
= ( 2x - 1 )( x2 + 3x - 4 )
= ( 2x - 1 )( x2 - x + 4x - 4 )
= ( 2x - 1 )[ x( x - 1 ) + 4( x - 1 ) ]
= ( 2x - 1 )( x - 1 )( x + 4 )
4) a2 + x2 - 16 + 2ax
= ( a2 + 2ax + x2 ) - 16
= ( a + x )2 - 42
= ( a + x - 4 )( a + x + 4 )