Factorize (20 marks)

1.25a^2 - 9b^2
=(5a+3b)(5a-3b)
2.x^4 - 16
=(x^2+4)(x^2-4)
=(x^2+4)(x+2)(x-2)
3.(6x-5)(x-y) - x(x-y)
=(x-y)(6x-5-x)
=(x-y)(5x-5)
=5(x-y)(x-1)
4.(2x+3)(x-y) + (1-x)(y-x)
=(2x+3)(x-y) - (1-x)(x-y)
=(x-y)(2x+3-x+y)
=(x-y)(3x+y+3)
5.p^2 - 4q^2 - p + 2q
=(p-2q)(p+2q)-(p-2q)
=(p-2q)(p+2q-1)
6.xy - 4x - 3y + 12
=x(y-4)-3(y-4)
=(y-4)(x-3)
7.ab + bc - ad - cd
=b(a+c)-d(a+c)
=(a+c)(b-d)
8. (3m+2n)^2 - (2m+n)^2
=9m^2+12mn+4n^2-4m^2-4mn-n^2
=5m^2+8mn+3n^2
=(5m+n)(m+3n)

25a^2 - 9b^2

= ( 5a )^2 - ( 3b )^2

= ( 5a - 3b ) ( 5a + 3b )


o o o o o o o o o o o o o o o


x^4 - 16

= ( x^2 )^2 - 4^2

= ( x^2 - 4 ) ( x^2 + 4 )

= ( x^2 - 2^2 ) ( x^2 + 4 )

= ( x + 2 ) ( x - 2 ) ( x^2 + 4 )


o o o o o o o o o o o o o o o


( 6x - 5 ) ( x - y ) - x ( x - y )

= ( 6x - 5 - x ) ( x - y )

= ( 5x - 5 ) ( x - y )

= 5 ( x - 1 ) ( x - y )


o o o o o o o o o o o o o o o


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

= 2x^2 - 2xy + 3x - 3y + y - x - xy + x^2

= 3x^2 -3xy + 2x - 2y

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


o o o o o o o o o o o o o o o


p^2 - 4q^2 - p + 2q

= p^2 - ( 2q )^2 - p + 2q

= ( p - 2q ) ( p + 2q ) - p + 2q

= ( p - 2q ) ( p + 2q - 1 )


o o o o o o o o o o o o o o o


xy - 4x - 3y + 12

= ( x - 3 ) ( y - 4 )


o o o o o o o o o o o o o o o


ab + bc - ad - cd

= b ( a + c ) - d ( a + c )

= ( b - d ) ( a + c )


o o o o o o o o o o o o o o o


( 3m + 2n )^2 - ( 2m + n )^2

= 9m^2 + 12mn + 4n^2 - 4m^2 - 4 mn - n^2

= 5m^2 + 8mn + 3n^2

= ( 5m + 3n ) ( m + n )

o o o o o o o o o o o o o o o

(1)
25a^2 - 9b^2
= 5^2a^2 -3^2b^2
=(5a+3b) (5a-3b)

(2)
x^4 - 16
=(x^2)^2 -4^2
=(x^2+4)(x^2-2^2)
=(x^2+4)(x+2)(x-2)

(3)
(6x-5)(x-y) - x(x-y)
=(x-y)(6x-5-x)
=(x-y)(5x-5)
=5(x-y)(x-1)

(4)
(2x+3)(x-y) + (1-x)(y-x)
=(2x+3)(x-y)+(-(x-y))(1-x)
=(2x+3)(x-y)+(x-y)(x-1)
=(x-y)(2x+3+x-1)
=(x-y)(3x+2)

(5)
p^2 - 4q^2 - p + 2q
=(p^2 -4q^2)-p+2q
=(p+2q)(p-2q) - (p-2q)
=(p-2q)(p+2q-1)

(6)
xy - 4x - 3y + 12
=(xy-4x)-(3y-12)
=x(y-4) -3(y-4)
=(y-4)(x-3)

(7)
ab + bc - ad - cd
=(ab+bc)-(ad+cd)
=b(a+c) -d(a+c)
=(b-d)(a+c)

(8)
(3m+2n)^2 - (2m+n)^2
=(3m+2n+2m+n)(3m+2n-(2m+n))
=(5m+3n)(3m+2n-2m-n)
=(5m+3n)(m+n)

1. 25a^2 - 9b^2 = (5a+3b)(5a-3b)
2. x^4 - 16 = (x^2+4)(x^2-4) = (x^2+4)(x+2)(x-2)
3. (6x-5)(x-y) - x(x-y) = (6x-5-x)(x-y) = (5x-5)(x-y) = 5(x-1)(x-y)
4. (2x+3)(x-y) + (1-x)(y-x) = (2x+3)(x-y) - (1-x)(x-y) =(2x+3-1+x)(x-y) = (3x+2)(x-y)
5. p^2 - 4q^2 - p + 2q = (p+2q)(p-2q) - (p-2q) = (p+2q-1)(p-2q)
6. xy - 4x - 3y + 12 = xy - 3y - 4x + 12 = y(x-3) - 4(x-3) = (y-4)(x-3)
7. ab + bc - ad - cd = b(a+c) - d(a+c) = (b-d)(a+c)
8. (3m+2n)^2 - (2m+n)^2 = (3m+2n+2m+n)(3m+2n-2m-n) = (5m+3n)(m+n)