Factorise the following by regrouping the terms.
a) 6xy-y^2+12xz-2yz
b) b^2-ab(1-a)-a^3
c) a^2-a(x+2y)+2xy
d) x^3-2x^2y+3xy^2-6y^3
e) xy(a^2+1)-a(x^2+y^2)
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Answers
Factorise the following by regrouping the terms.
a) 6xy - y² + 12xz - 2yz
b) b² - ab(1 - a) - a³
c) a² - a(x + 2y) + 2xy
d) x³ - 2x²y + 3xy² - 6y³
e) xy(a² + 1)-a(x² + y²)
a)
◘ The factorised form is (6x - y)(y + 2z).
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b)
◘ The factorised form is (b - a)(b + a²).
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c)
◘ The factorised form is (a - x)(a - 2y).
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d)
◘ The factorised form is (x - 2y)(x² + 3y²).
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e)
◘ The factorised form is (ax - y)(ay - x).
Answer:
Solution :
[ a ]
6xy - y² + 12xz - 2yz
=> y ( 6x - y ) - 12z ( 6x - y )
=> ( 6x - y )( y - 12z )
[ b ]
b² - ab ( 1 - a ) - a³
=> b² - ab + a²b - a³
=> b ( b - a ) + a²( b - a )
=> ( a² + b )( b - a )
[ c ]
a² - a ( x + 2y ) + 2xy
=> a² - ax - 2ay + 2xy
=> a( a - x ) - 2y ( a - x )
=> ( a - x )( a - 2y )
[ d ]
x³ - 2x²y + 3xy² - 6y³
=> x² ( x - 2y ) + 3y² ( x - 2y )
=> ( x² + 3y² )( x - 2y )
[ e ]
xy ( a² + 1 ) - a ( x² + y² )
=> a²xy + xy - ax² - ay²
=> a²xy - ax² + xy - ay²
=> ax ( ay - x ) - y ( ay - x )
=> ( ax - y )( ay - x ) .
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Additional Information -
( a + b )² = a² + 2ab + b²
( a - b )² = a² - 2ab + b²
( a + b )( a - b ) = a² - b²
( a + b )³ = a³ + 3ab ( a + b ) + b³
( a - b )³ = a³ - 3ab ( a + b ) - b³
( a + b + c )³ = a³ + b³ + c³ + 3 ( a + b )( b + c )( c + a )
a³ + b³ + c³ - 3abc = ( a + b + c )( a² + b² + c² - ab - bc - ca )
When a + b + c = 0 ,
a³ + b³ + c³ = 3abc .
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