a) 5x3 - 40 = 5( x3 - 8 ) = 5( x - 2 )( x2 + 2x + 4 )
b) x2z + 4xyz + 4y2z = z( x2 + 4xy + 4y2 ) = z( x + 2y )2
c) 4x2 - y2 - 6x + 3y = ( 4x2 - y2 ) - ( 6x - 3y ) = ( 2x - y )( 2x + y ) - 3( 2x - y ) = ( 2x - y )( 2x + y - 3 )
d) x2 + 2x - 4y2 + 1 = ( x2 + 2x + 1 ) - 4y2 = ( x + 1 )2 - ( 2y )2 = ( x - 2y + 1 )( x + 2y + 1 )
e) 3x2 - 3y2 - 12x + 12y = 3( x2 - y2 - 4x + 4y ) = 3[ ( x2 - y2 ) - ( 4x - 4y ) ] = 3[ ( x - y )( x + y ) - 4( x - y ) ] = 3( x - y )( x + y - 4 )
f) x3 + 5x2 + 4x + 20 = x2( x + 5 ) + 4( x + 5 ) = ( x + 5 )( x2 + 4 )
g) x3 - x2 - 25x + 25 = x2( x - 1 ) - 25( x - 1 ) = ( x - 1 )( x2 - 25 ) = ( x - 1 )( x - 5 )( x + 5 )
a) \(5x^3-40=5\left(x^3-8\right)=5\left(x-2\right)\left(x^2+2x+4\right)\)
b) \(x^2z+4xyz+4y^2z=z\left(x^2+4xy+4y^2\right)=z\left(x+2y\right)^2\)
c) \(4x^2-y^2-6x+3y=\left(4x^2-y^2\right)-\left(6x-3y\right)\)
\(=\left(2x-y\right)\left(2x+y\right)-3\left(2x-y\right)=\left(2x-y\right)\left(2x+y-3\right)\)
d) \(x^2+2x-4y^2+1=x^2+2x+1-4y^2\)
\(=\left(x+1\right)^2-4y^2=\left(x+2y+1\right)\left(x-2y+1\right)\)
e) \(3x^2-3y^2-12x+12y=3\left(x^2-y^2-4x+4y\right)\)
\(=3\left[\left(x^2-y^2\right)-\left(4x-4y\right)\right]=3\left[\left(x-y\right)\left(x+y\right)-4\left(x-y\right)\right]\)
\(=3\left(x-y\right)\left(x+y+4\right)\)
f) \(x^3+5x^2+4x+20=\left(x^3+5x^2\right)+\left(4x+20\right)\)
\(=x^2.\left(x+5\right)+4\left(x+5\right)=\left(x^2+4\right)\left(x+5\right)\)
g) \(x^3-x^2-25x+25=\left(x^3-x^2\right)-\left(25x-25\right)\)
\(=x^2\left(x-1\right)-25\left(x-1\right)=\left(x-1\right)\left(x^2-25\right)\)
\(=\left(x-1\right)\left(x-5\right)\left(x+5\right)\)
a, \(5x^3-40=5\left(x^3-8\right)=5\left(x^3-2^3\right)=5\left(x-2\right)\left(x^2+2x+4\right)\)
b, \(x^2z+4xyz+4y^2z=z\left(x^2-4xy+4y^2\right)=z\left(x-2y\right)^2\)
c, \(4x^2-y^2-6x+3y=\left(2x-y\right)\left(2x+y\right)-3\left(2x+y\right)=\left(2x-y-3\right)\left(2x+y\right)\)
d, \(x^2+2x-4y^2+1=\left(x+1\right)^2-4y^2=\left(x+1-4y\right)\left(x+1+4y\right)\)
e, \(3x^2-3y^2-12x+12y=3\left(x-y\right)\left(x+y\right)-12\left(x-y\right)=\left(x-y\right)\left(3x+3y-12\right)\)
f, \(x^3+5x^2+4x+20=x^2\left(x+5\right)+4\left(x+5\right)=\left(x^2+4\right)\left(x+5\right)\)
g, \(x^3-x^2-25x+25=x^2\left(x-1\right)-25\left(x-1\right)=\left(x-5\right)\left(x+5\right)\left(x-1\right)\)
a) \(5x^3-40=5.\left(x^3-8\right)\)
b) \(x^2z+4xyz+4y^2z=z.\left(x^2+4xy+4y^2\right)=z.\left(x+2y\right)^2\)
c) \(4x^2-y^2-6x+3y=\left(4x^2-y^2\right)-\left(6x-3y\right)=\left(2x-y\right).\left(2x+y\right)-3.\left(2x-y\right)\)
\(=\left(2x-y\right).\left(2x+y-3\right)\)
d) \(x^2+2x-4y^2+1=\left(x^2+2x+1\right)-4y^2=\left(x+1\right)^2-\left(2y\right)^2\)
\(=\left(x+1-2y\right).\left(x+1+2y\right)\)
e) \(3x^2-3y^2-12x+12y=\left(3x^2-3y^2\right)-\left(12x-12y\right)=3.\left(x^2-y^2\right)-12.\left(x-y\right)\)
\(=3.\left(x+y\right).\left(x-y\right)-12.\left(x-y\right)=\left(x-y\right).\left[3.\left(x+y\right)-12\right]=\left(x-y\right).3.\left(x+y-4\right)\)
\(=3.\left(x-y\right).\left(x+y-4\right)\)
f) \(x^3+5x^2+4x+20=\left(x^3+5x^2\right)+\left(4x+20\right)=x^2.\left(x+5\right)+4.\left(x+5\right)\)
\(=\left(x+5\right).\left(x^2+4\right)\)
g) \(x^3-x^2-25x+25=\left(x^3-x^2\right)-\left(25x-25\right)=x^2.\left(x-1\right)-25.\left(x-1\right)\)
\(=\left(x-1\right).\left(x^2-25\right)=\left(x-1\right).\left(x-5\right).\left(x+5\right)\)
a.\(5x^3-40\)
\(=5\left(x^3-8\right)\)
\(=5\left(x^3-2^3\right)\)
\(=5\left(x-2\right)\left(x^2+2x+4\right)\)
b,\(x^2z+4xyz+4y^2z\)
\(=z\left(x^2+4xy+4y^2\right)\)
\(=z\left(x+2y\right)^2\)
c,\(4x^2-y^2-6x+3y\)
d,\(x^2+2x-4y^2+1\)
\(=\left(x^2+2x+1\right)-\left(2y\right)^2\)
\(=\left(x+1\right)^2-\left(2y\right)^2\)
\(=\left(x+1-2y\right)\left(x+1+2y\right)\)
e\(3x^2-3y^2-12x+12y\)
\(=3\left(x^2-y^2\right)-12\left(x-y\right)\)
\(=3\left(x-y\right)\left(x+y\right)-12\left(x-y\right)\)
\(=\left(x-y\right)\left[3\left(x+y\right)-12\right]\)
