Rational LCD?

Question

x/ x-3 - 2/x+4 - 14/x^2 + x-12

OK the lowest common demometer is x^2 +x -12 (I think)
how do I get that across the board though?

Answer 1

You multiply the first term by (x+4)/(x+4), and the second term by (x-3)/(x-3).

x/(x-3) (x+4)/(x+4) - 2/(x+4) (x+3)/(x+3) - 14 / (x^2 + x -12)

[x(x+4) - 2(x+3) - 14] / (x^2 + x - 12)

This is the algebraic equivalent of something like:

1/3 + 1/4 + 1/12
1/3 (4/4) + 1/4 (3/3) + 1/12
4/12 + 3/12 + 1/12 = (4+3+1)/12 = 8/12 = 2/3

Answer 2

x/ x-3 - 2/x+4 - 14/x^2 + x-12

the denominators are
x-3,
x+ 4,
x^2 + x - 12 = (x+4)(X-3)

now multiply all terms by (x+4)(X-3) and still use it as your new denminator
x/ x-3 - 2/x+4 - 14/x^2 + x-12

(x+4)(X-3)x/ x-3 - (x+4)(X-3)2/x+4 - x+4)(X-3)14/x+4)(X-3)
______________________________________________
(x+4)(X-3)

cancel all terms that are similar in the numerator

(x+4)x - (X-3)2 - 14
________________
(x+4)(X-3)

simplify

x^2 + 4X - 2x + 6 -14
__________________
(x+4)(X-3)

simplify further

x^2 -2x - 8
__________
(x+4)(X-3)

but x^2 -2x - 8 can be factored as (x-4)(x+2)

(x-4)(x+2)
__________
(x+4)(X-3)