F.3 Factorization

1)
Let a = 3x - 2
and b = x + 1

(3x - 2)3 - 8(x + 1)3

= a3 - 8b3

= a3 - (2b)3

= (a - 2b)[a2 + a(2b) + (2b)2]

= [(3x - 2) - 2(x + 1)][(3x - 2)2 + (3x - 2)2(x + 1) + 22(x + 1)2]

= (3x - 2 - 2x - 2)(9x2 - 12x + 4 + 6x2 + 2x - 4 + 4x2 + 8x + 4)

= (x - 4)(19x2 - 2x + 4)

=====
2)
Let x = a + b
and y = a - b

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

= x3 + y3

= (x + y)(x2 + xy + y2)

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

= (a + b + a - b)(a2 + 2ab + b2 - a2 + b2 + a2 -2ab + b2)

= 2a(a2 + 3b2)

=====
3)
Let a = x + 2
and b = y - 3

(x + 2)2 - (x + 2)(y - 3) - 6(y - 3)2

= a2 - ab - 6b2

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

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

= [x + 2 - 3y + 9][x + 2 + 2y - 6)]

= (x - 3y + 11)(x + 2y - 4)

=====
4)
Let a = x + 7
and b = x - 6

3(x+7)2 + (x+7)(x-6) - 14(x-6)2

= 3a2 + ab - 14b2

= (3a + 7b)(a - 2b)

= [3(x + 7) + 7(x - 6)][(x + 7) - 2(x - 6)]

= [3x + 21 + 7x - 42][x + 7 - 2x + 12]

= (10x - 21)(-x + 19)

= (10x - 21)(19 - x)
=