[simplifying rational expressions][section 52]

(1.)    6x + 12
__________________                  here is the problem
x2 - x - 6

=     6(x + 2)
_____________________
(x + 2)(x - 3)                    factor

=   6/(x - 3)               cancel

(2.)     x2y - y
___________                 here is the problem
x + 1

=     y(x2 - 1)
_______________
x + 1                           factor

=   y(x - 1)(x + 1)
_________________
(x + 1)                    factor

=     y(x - 1)               cancel

(3.)  3a2 - 3
__________                   here is the problem
1 - a

=     3(a2 - 1)
__________
1 - a                factor

=   3(a - 1)(a + 1)
_________________
-(a - 1)                factor

=   -3(a + 1)       cancel

(4.)   a2 + 1
_________             here is the problem
a + 1

[cannot be simplified any further]

(5.)   6a2 + a - 15
___________________                here is the problem
6a2 - 13a + 6

=   (3a + 5)(2a - 3)
______________________           factor
(3a - 2)(2a - 3)

=    3a + 5
______
3a - 2                     cancel

(6.)   2a2 - ab - b2
_____________________          here is the problem
2a2 + ab - b2

=     (2a + b)(a - b)
_____________________         factor
(2a - b)(a + b)

(7.)    6a2 + 42a + 72
___________________              here is the problem
18 - 2a2

=     6(a2 + 7a + 12)
______________________
2(9 - a2)                   factor

=     6(a + 3)(a + 4)
_______________________
2(3 - a)(3 + a)           factor

=    -3(a + 4)
________________           cancel
(a - 3)

(8.)    3x2 - 27
____________________                here is the problem
24 - 11x + x2

=    3(x2 - 9)
____________________
(x - 8)(x - 3)                    factor

=   3(x - 3)(x + 3)
____________________
(x - 3)(x - 8)                factor

=    3(x + 3)
_____________
(x - 8)                cancel

(9.)     (a + b)2(a - b)
________________________          here is the problem
a2 - b2

=   (a + b)2(a - b)
___________________
(a + b)(a - b)                   factor

=  a + b                       cancel

(10.)    x2 - y2
___________________          here is the problem
x2 - 2xy + y2

=    (x - y)(x + y)
______________________
(x - y)(x - y)                 factor

=   (x + y)/(x - y)              cancel