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a: \(x^4+3x^3+x^2+3x\)
\(=x\left(x^3+3x^2+x+3\right)\)
\(=x\left(x+3\right)\left(x^2+1\right)\)
c: \(x^2-xy-x+y\)
\(=x\left(x-y\right)-\left(x-y\right)\)
\(=\left(x-y\right)\left(x-1\right)\)
a) \(x^2-5xy+6y^2\)
\(=x^2-3xy-2xy+6y^2\)
\(=x\left(x-3y\right)-2y\left(x-3y\right)\)
\(=\left(x-2y\right)\left(x-3y\right)\)
b) \(16\left(x-1\right)^2-36y^2\)
\(=\left(4x-4\right)^2-\left(6y\right)^2\)
\(=\left(4x+6y-4\right)\left(4x-6y-4\right)\)
c) \(4\left(x+y\right)-12\left(x+y\right)^2\)
\(=\left(x+y\right)\left[4-12\left(x+y\right)\right]\)
\(=4\left(x+y\right)\left[1-3x-3y\right]\)
\(1,=3xy\left(x^2+2xy+y^2\right)=3xy\left(x+y\right)^2\\ 2,=7xy\left(2x-3y+4xy\right)\\ 3,=\left(x-1\right)\left(x^2-4\right)=\left(x-2\right)\left(x+2\right)\left(x-1\right)\\ 4,=\left(x-y\right)\left(10x+8\right)=2\left(5x+4\right)\left(x-y\right)\\ 5,=\left(b-c\right)\left(8a-6b\right)=2\left(4a-3b\right)\left(b-c\right)\\ 6,=\left(x-1\right)\left(x^2-16\right)=\left(x-4\right)\left(x+4\right)\left(x-1\right)\\ 7,=x\left(x-y\right)+5\left(x-y\right)=\left(x-y\right)\left(x+5\right)\\ 8,=\left(3x+1-x-1\right)\left(3x+1+x+1\right)=2x\left(4x+2\right)=4x\left(2x+1\right)\\ 9,=\left(2x-3y\right)\left(4x^2+6xy+9y^2\right)\\ 10,=\left(x-1\right)^2-4y^2=\left(x-2y-1\right)\left(x+2y-1\right)\)
\(a,4\left(2-x\right)^2+xy-2y\)
\(=4\left(2-x\right)^2-y\left(2-x\right)\)
\(=4-y\left(2-x\right)^2\left(2-x\right)\)
\(=\left(2-x\right)\left[\left(2-x\right)4-y\right]\)
\(=\left(2-x\right)\left(4x-8+y\right)\)
\(c,x^3+y^3+z^3-3xyz\)
\(=x^3+y^3+z^3+3x^2y-3x^2y+3xy^2-3xy^2-3xyz\)
\(=\left(x^3+3x^2y+3xy^2+y^3\right)-3x^2y-3xy^2+z^3-3xyz\)
\(=\left(x+y\right)^3-3xy\left(x+1\right)+z^3-3xyz\)
\(=\left[\left(x+y\right)^3+z^3\right]-3xy\left(x+y\right)-3xyz\)
\(=\left[\left(x+y\right)+z\right]\left[\left(x+y\right)^2-\left(x+y\right)z+z^2\right]-3xy\left(x+y+z\right)\)
\(=\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2\right)-3xy\left(x+y+z\right)\)
\(=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-xz-yz\right)\)
a) 4(2 - x)2 + xy - 2y = 4(x - 2)2 + y(x - 2) = (4x - 8 + y)(x - 2)
b) 2(x - 1)3 - 5(x - 1)2 - (x - 1) = (x - 1)[2(x - 1)2 - 5(x - 1) - 1]
= (x - 1)(2x2 - 4x + 2 - 5x + 5 - 1) = (x - 1)(2x2 - 9x + 6)
c) x3 + y3 + z3 - 3xyz = (x + y)(x2 - xy + y2) + z3 - 3xyz
= (x + y)3 + z3 - 3xy(x + y) - 3xyz = (x + y + z)(x2 + 2xy + y2 - xz - yz + z2) - 3xy(x + y + z)
= (x + y + z)(x2 + y2 + z2 - xz - yz + 2xy - 3xy) = (x + y + z)(x2 + y2 + z2 - xy - yz - xz)
a) x^2-9x+20
= x^2+4x+5x+20
=(x^2+4x)+(5x+20)
=x(x+4)+5(x+4)
=(x+4)(x+5)
2: \(8xy-24xy+16x\)
\(=8x\cdot y-8x\cdot3y+8x\cdot2\)
\(=8x\left(y-3y+2\right)=8x\left(-2y+2\right)\)
\(=-16y\left(y-1\right)\)
3: \(xy-x=x\cdot y-x\cdot1=x\left(y-1\right)\)
11: \(2mx-4m2xy+6mx\)
\(=2mx-2my\cdot4y+2mx\cdot3\)
\(=2mx\left(1-4y+3\right)\)
\(=2mx\left(4-4y\right)=8mx\left(1-y\right)\)
12: \(7x^2y^5-14x^3y^4-21y^3\)
\(=7y^3\cdot x^2y^2-7y^3\cdot2x^3y-7y^3\cdot3\)
\(=7y^3\left(x^2y^2-2x^3y-3\right)\)
13: \(2\left(x-y\right)-a\left(x-y\right)\)
\(=2\cdot\left(x-y\right)-a\cdot\left(x-y\right)\)
\(=\left(x-y\right)\left(2-a\right)\)
a, x2+2x+1+x+1
=(x2+2x+2)+x
=(x2+2x+12)+x
=(x+1)2+x
=(2x+1)2
=(2x-1).(2x+1 )
c,xy-y-2x-2
=(xy-2x)-(y-2)
=x.(y-2)-(y-2)
=(y-2).x
e,xy+xz+y2+yz
=(xy+y2)+(xz+yz)
=y.(x+y)+z.(x+y)
=(x+y).(y+z)
d,x3+x2+x+1
=(x3+x2)+(x+1)
=x2.(x+1)+(x+1)
=x2.(x+1)
b,y2+xy+x+2y+1
=(y2+2y)+(xy+x+1)
=y.(y+2) + x.(y+2)
=(y+2).(y+x)
