a, \(\left(a-b\right)\left(b-a\right)y+a-b\)

\(=\left(a-b\right)\left(by-ay+1\right)\)

b, \(\left(x-y+z\right)\left(a+y-x-z\right)b-x+y-z\)

\(=\left(x-y+z\right)\left(a+y-x-z\right)b-\left(x-y+z\right)\)

\(=\left(x-y+z\right)\left(ab+yb-xb-xz-1\right)\)

c, \(\left(2a+3\right)x-\left(2a+3\right)y+2a+3\)

\(=\left(2a+3\right)\left(x-y\right)+\left(2a+3\right)\)

\(=\left(2a+3\right)\left(x-y+1\right)\)

d, \(\left(a-b\right)x+\left(b-a\right)y-a+b\)

\(=\left(a-b\right)x-\left(a-b\right)y-\left(a-b\right)\)

\(=\left(a-b\right)\left(x-y-1\right)\)

a. (a-b)(b-a)y+a-b=(a-b)[(b-a)y+1]

b. (x-y+z)(a+y-x-z)b-x+y-z=(x-y+z)(a-y-x-z)b-(x-y+z)=(x-y+z)[(a-y-x-z)b-1]

c. (2a+3)x-(2a+3)y+2a+3=(2a+3)(x-y+1)

d. (a-b)x+(b-a)y-a+b=(a-b)x-(a-b)y-(a-b)=(a-b)(x-y-1)

a)x+2a.(x-y)-y=(x-y)+2a(x-y)

=(x-y)(1+2a)

b)x^2-(a+b)x+ab=[x^2-(a+b)x]+a

=x​(x-a-b)+a

a,\(\left(a-b\right)\left(a+2b\right)-\left(b-a\right)\left(2a-b\right)-\left(a-b\right)\left(a+3b\right)\)

\(=\left(a-b\right)\left(a+2b\right)+\left(a-b\right)\left(2a-b\right)-\left(a-b\right)\left(a+3b\right)\)

\(=\left(a-b\right)\left(a+2b+2a-b-a-3b\right)\)

\(=\left(a-b\right)\left(2a-2b\right)\)

\(=\left(a-b\right)2\left(a-b\right)\)

\(=2\left(a-b\right)^2\)

b,\(\left(x+y\right)\left(2x-y\right)+\left(2x-y\right)\left(3x-y\right)-\left(y-2x\right)\)

\(=\left(x+y\right)\left(2x-y\right)+\left(2x-y\right)\left(3x-y\right)+\left(2x-y\right)\)

\(=\left(2x-y\right)\left(x+y+3x-y+1\right)\)

\(=\left(2x-y\right)\left(4x+1\right)\)

c,\(x^2\left(y-z\right)+y^2\left(z-x\right)+z^2\left(x-y\right)\)

\(=x^2y-x^2z+y^2z-y^2x+z^2\left(x-y\right)\)

\(=x^2y-y^2x-x^2z+y^2z+z^2\left(x-y\right)\)

\(=xy\left(x-y\right)-z\left(x^2-y^2\right)+z^2\left(x-y\right)\)

\(=xy\left(x-y\right)-z\left(x-y\right)\left(x+y\right)+z^2\left(x-y\right)\)

\(=\left(x-y\right)\left(xy-zx-zy+z^2\right)\)

\(=\left(x-y\right)\left(y-z\right)\left(x-z\right)\)

a) ( a - b )( a + 2b ) - ( b - a )( 2a - b ) - ( a - b )( a + 3b )

= ( a - b )( a + 2b ) + ( a - b )( 2a - b ) - ( a - b )( a + 3b )

= ( a - b )[ ( a + 2b ) + ( 2a - b ) - ( a + 3b ) ]

= ( a - b )( a + 2b + 2a - b - a - 3b )

= ( a - b )( 2a - 2b )

= 2( a - b )( a - b ) = 2( a - b )²

b) ( x + y )( 2x - y ) + ( 2x - y )( 3x - y ) - ( y - 2x )

= ( x + y )( 2x - y ) + ( 2x - y )( 3x - y ) + ( 2x - y )

= ( 2x - y )[ ( x + y ) + ( 3x - y ) + 1 ]

= ( 2x - y )( x + y + 3x - y + 1 )

= ( 2x - y )( 4x + 1 )

c) x²( y - z ) + y²( z - x ) + z²( x - y )

= x²y - x²z + y²z - y²x + z²( x - y )

= ( x²y - y²x ) - ( x²z - y²z ) + z²( x - y )

= xy( x - y ) - z( x² - y² ) + z²( x - y )

= xy( x - y ) - z( x - y )( x + y ) + z²( x - y )

= ( x - y )[ xy - z( x + y ) + z² ]

= ( x - y )( xy - zx - zy + z² )

= ( x - y )[ ( xy - zx ) - ( zy - z² ) ]

= ( x - y )[ x( y - z ) - z( y - z ) ]

= ( x - y )( y - z )( x - z )

= (x+y)-(x-y).2a

=-(x-y)-(x-y).2a

=(x-y).2a

a x+ 2a(x-y)-y

=x-y+2a(x-y)

=(x-y)(1+2a)

b x^2-(a+b)x+ab

=x^2-(xa+xb)+ab

=x^2-xa-xb+ab

=x(x-a)-b(x-a)

=(x-a)(x-b)

f x(y^2-z^2)+y(z^2-x^2)+z(x^2-y^2)

=xy^2-xz^2+yz^2-yx^2+zx^2-zy^2

=-xzy^2-xyz^2-zyx^2

=-xyz(y+z+x)

2

a

\(x+y+z=0\)

\(\Rightarrow x+y=-z\)

\(\Rightarrow\left(x+y\right)^3=\left(-z\right)^3\)

\(\Rightarrow x^3+y^3+3x^2y+3xy^2=-z^3\)

\(\Rightarrow x^3+y^3+z^3=3xy\left(x+y\right)=3xyz\)

b

Đặt \(a-b=x;b-c=y;c-a=z\Rightarrow x+y+z=0\)

Ta có bài toán mới:Cho \(x+y+z=0\).Phân tích đa thức thành nhân tử:\(x^3+y^3+z^3\)

Áp dụng kết quả câu a ta được:

\(\left(a-b\right)^3+\left(b-c\right)^3+\left(c-a\right)^3=3\left(a-b\right)\left(b-c\right)\left(c-a\right)\)

\(\left(2a+3\right)\left(2a+3\right)y+\left(2a+3\right)\)

\(=\left(2a+3\right)[y\left(2a+3\right)+1]\)

\(=\left(2a+3\right)\left(2ay+3y+1\right)\)

\(\left(a-b\right)x+\left(b-a\right)y-\left(a-b\right)\) (Sửa đề)

\(=\left(a-b\right)x-\left(a-b\right)y-\left(a-b\right)\)

\(=\left(a-b\right)\left(x-y-1\right)\)

\(\left(4x-y\right)\left(a+b\right)+\left(4x-y\right)\left(c-1\right)\)

\(=\left(4x-y\right)[\left(a+b\right)+\left(c-1\right)]\)

\(=\left(4x-y\right)\left(a+b+c-1\right)\)