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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
a) \(x^3-4x^2+5x-2\)
=\(x^3-x^2-3x^2+3x+2x-2\)
=\(x^2\left(x-1\right)-3x\left(x-1\right)+2\left(x-1\right)\)
=\(\left(x-1\right)\left(x^2-3x+2\right)\)
=\(\left(x-1\right)\left(x^2-x-2x-2\right)\)
=\(\left(x-1\right)\left(\left(x\left(x-1\right)-2\left(x-1\right)\right)\right.\)
=\(\left(x-1\right)^2\left(x-2\right)\)
b) \(\) \(x^5+x+1\)
=\(x^5-x^2+x^2+x+1\)
=\(x^2\left(x^3-1\right)+\left(x^2+x+1\right)\)
=\(x^2\left(x-1\right)\left(x^2+x+1\right)+\left(x^2+x+1\right)\)
=\(\left(x^2+x+1\right)\left(x^3-x^2+1\right)\)
c) \(x^3+5x^2+5x+1\)
=\(\left(x^3+1\right)+\left(5x^2+5x\right)\)
=\(\left(x+1\right)\left(x^2-x+1\right)+5x\left(x+1\right)\)
=\(\left(x+1\right)\left(x^2+4x+1\right)\)
d) \(x^2\left(x^2+2y^2\right)-3y^4\)
=\(x^4+2x^2y^2-3y^4\)
=\(x^4-x^2y^2+3x^2y^2-3y^4\)
=\(x^2\left(x^2-y^2\right)+3y^2\left(x^2-y^2\right)\)
=\(\left(x^2-y^2\right)\left(x^2+3y^2\right)\)
=\(\left(x-y\right)\left(x+y\right)\left(x^2+3y^2\right)\)
\(4x^2-25+\left(2x+7\right).\left(5-2x\right)\)
\(=\left(2x+5\right).\left(2x-5\right)-\left(2x+7\right).\left(2x-5\right)\)
\(=\left(2x+5-2x-7\right).\left(2x-5\right)\)
\(=-2.\left(2x-5\right)\)
\(a^2x^2-a^2x^2-b^2x^2+b^2y^2\)
\(=a^2.\left(x^2-y^2\right)-b^2.\left(x^2-y^2\right)\)
\(=\left(a^2-b^2\right).\left(x^2-y^2\right)\)
\(=\left(a-b\right).\left(a+b\right).\left(x-y\right).\left(x+y\right)\)
\(x^2-y^2+12y-36\)
\(=x^2-\left(y^2-12y+36\right)\)
\(=x^2-\left(y-6\right)^2\)
\(=\left(x-y+6\right).\left(x+y-6\right)\)
\(\left(x+2\right)^2-x^2+2x-1\)
\(=\left(x+2\right)^2-\left(x^2-2x+1\right)\)
\(=\left(x+2\right)^2-\left(x-1\right)^2\)
\(=[x+2-\left(x-1\right)].[x+2+\left(x-1\right)]\)
\(=\left(x+2-x+1\right).\left(x+2+x-1\right)\)
\(=3.\left(2x+1\right)\)
\(16x^2-y^2=\left(4x\right)^2-y^2=\left(4x-y\right).\left(4x+y\right)\)
\(1+27x^3=1^3+\left(3x\right)^3=\left(1+3x\right).\left(1-3x+9x^2\right)\)