[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