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a) = 5( x2 - 9y2 - 6y - 1 ) = 5[ x2 - ( 9y2 + 6y + 1 ) ] = 5[ x2 - ( 3y + 1 )2 ] = 5( x - 3y - 1 )( x + 3y + 1 )
b) = 125x3 - 25x2 + 15x2 - 3x + 5x - 1 = 25x2( 5x - 1 ) + 3x( 5x - 1 ) + ( 5x - 1 ) = ( 5x - 1 )( 25x2 + 3x + 1 )
c) = 5( x - 7 ) + a( x - 7 ) = ( x - 7 )( a + 5 )
d) = ( a - b )2 + ( a - b ) = ( a - b )( a - b + 1 )
e) = ax2 + a - a2x - x = ax( a - x ) + ( a - x ) = ( a - x )( ax + 1 )
f) = ( 10x )2 - ( x2 + 25 )2 = ( 10x - x2 - 25 )( 10x + x2 + 25 ) = -( x - 5 )2( x + 5 )2
g) \(x^5-3x^4+3x^3-x^2=x^2\left(x^3-3x^2+3x-1\right)=x^2\left(x-1\right)^3\)
f) \(x^2-25-2xy+y^2=\left(x^2-2xy+y^2\right)-25=\left(x-y\right)^2-5^2=\left(x-y-5\right)\left(x-y+5\right)\)
e) \(16x^3+54y^3=2\left(8x^3+27y^3\right)=2\left[\left(2x\right)^3+\left(3y\right)^3\right]=2\left(2x+3y\right)\left(4x^2-6xy+9y^2\right)\)
d) \(3y^2-3z^2+3x^2+6xy=3\left(x^2+2xy+y^2-z^2\right)=3\left[\left(x+y\right)^2-z^2\right]=3\left(x+y+z\right)\left(x+y-z\right)\)
\(4x^2-25+\left(2x+7\right).\left(5-2x\right)\)
\(=\left(2x+5\right).\left(2x-5\right)-\left(2x+7\right).\left(2x-5\right)\)
\(=\left(2x+5-2x-7\right).\left(2x-5\right)\)
\(=-2.\left(2x-5\right)\)
\(a^2x^2-a^2x^2-b^2x^2+b^2y^2\)
\(=a^2.\left(x^2-y^2\right)-b^2.\left(x^2-y^2\right)\)
\(=\left(a^2-b^2\right).\left(x^2-y^2\right)\)
\(=\left(a-b\right).\left(a+b\right).\left(x-y\right).\left(x+y\right)\)
\(x^2-y^2+12y-36\)
\(=x^2-\left(y^2-12y+36\right)\)
\(=x^2-\left(y-6\right)^2\)
\(=\left(x-y+6\right).\left(x+y-6\right)\)
\(\left(x+2\right)^2-x^2+2x-1\)
\(=\left(x+2\right)^2-\left(x^2-2x+1\right)\)
\(=\left(x+2\right)^2-\left(x-1\right)^2\)
\(=[x+2-\left(x-1\right)].[x+2+\left(x-1\right)]\)
\(=\left(x+2-x+1\right).\left(x+2+x-1\right)\)
\(=3.\left(2x+1\right)\)
\(16x^2-y^2=\left(4x\right)^2-y^2=\left(4x-y\right).\left(4x+y\right)\)
\(1+27x^3=1^3+\left(3x\right)^3=\left(1+3x\right).\left(1-3x+9x^2\right)\)
Answer:
\(5x^2-10xy+5y^2-20z^2\)
\(=5.\left(x^2-2xy+y^2-4z^2\right)\)
\(=5.[\left(x+y\right)^2-\left(2z\right)^2]\)
\(=5.\left(x+y-2z\right).\left(x+y+2z\right)\)
\(16x-5x^2-3\)
\(=\left(-5x^2+15x\right)+\left(x-3\right)\)
\(=-5x.\left(x-3\right)+\left(x-3\right)\)
\(=\left(1-5x\right).\left(x-3\right)\)
\(x^2-5x+5y-y^2\)
\(=(x-y).(x+y)-5.(x-y)\)
\(=(x-y).(x+y-5)\)
\(3x^2-6xy+3y^2-12z^2\)
\(=3.(x^2-2xy+y^2-4z^2)\)
\(=3[\left(x-y\right)^2-\left(2z\right)^2]\)
\(=3.(x-y-2z).(x-y+2z)\)
\(x^2+4x+3\)
\(=(x^2+x)+(3x+3)\)
\(=x.(x+1)+3.(x+1)\)
\(=(x+1).(x+3)\)
\((x^2+1)^2-4x^2\)
\(=(x^2-2x+1).(x^2+2x+1)\)
\(=(x-1)^2.(x+1)^2\)
\(x^2-4x-5\)
\(=(x^2+x)-(5x+5)\)
\(=x.(x+1)-5.(x+1)\)
\(=(x-5).(x+1)\)
a) = ( x - 3 )2 - ( 3x )2 = ( -2x - 3 )( 4x - 3 )
b) = x( x - 2y ) - ( x - 2y ) = ( x - 2y )( x - 1 )
a) \(\left(x-3-3x\right).\left(x-3+3x\right)\)
b) \(\left(x-2y\right).\left(x-1\right)\)
a) x2 - 2xy + 5x - 10y
= x(x - 2y) + 5(x - 2y)
= (x + 5)(x - 2y)
b) x(2x - 3y) - 6y2 + 4xy
= x(2x - 3y) + 2y(2x - 3y)
= (x + 2y)(2x - 3y)
c) 8x3 + 4x2 - y2 - y3
= (4x2 - y2) + (8x3 - y3)
= (2x - y)(2x + y) - (2x - y)(4x2 + 2xy + y2)
= (2x - y)(-4x2 - 2xy - y2 + 2x + y)
d) a3 - a2b - ab2 + b3
= a2(a- b) - b2(a - b)
= (a2 - b2)(a - b) = (a - b)2(a + b)
e) ab2c3 + 64ab2
= ab2(c3 + 64)
= ab2(c + 4)(c2 + 4c + 16)
f) 27x3y - a3b3y
= y[27 - (ab)3]
= y(3 - ab)(a2b2 + 3ab + 9)
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