In partial fractions, what is the full written sequence for 11-3X/(X-1)(X+3)?

Answer 1

I'm not entirely sure if you meant (11-3X) on the top, or 11 - (the rest). I'm going to assume the former - if not, hopefully you'll be able to work it out yourself with the same method.
We want to express (11-3X)/((X-1)(X+3)) as A/(X-1) + B/(X+3) for some A and B. There are a couple of ways of doing this.
The purely algebraic way would be the following:
First put everything over a common denominator: that gives a numerator of A(X+3) + B(X-1) = AX+BX+3A-B by crossmultiplying. So we want (A+B) = -3 to get the X part working, and 3A-B = 11 for the constant bit. Add those equations to get 4A = 8, so A = 2, and thus B = -5.
So its 2/(X-1) - 5/(X+3).

There is another way that works that you may find easier. 'Cover' the (X-1) part on the denominator with your hand. Now its perfectly fine to substitute X=1, since you can't see the divide-by-0 on the bottom. When you do that, you get 8/4 = 2.
Similarly, cover the X+3 part, and substitute X=-3. That gives 20/-4 = -5.
And those two numbers happen to be precisely the same numbers we came up with before. And that always works.

Answer 2

= 11-3X / X^2+2X-3

Answer 3

(11-3x)/(x-1)(x+3)
=A/(x-1)+B/(x+3)

multiply through by (x-1)(x+3)
(11-3x)=A(x+3)+B(x-1)
put x=1
8=4A
A=2
put x= -3
20= -4B
B= -5

therefore,
(11-3x)/(x-1)(x+3)
=2/(x-1)-5/(x+3)

i hope that this helps