Question

When you are decomposing fractions into constants

EX:

1/(x-1)(x-2)^2 = A/(x-1) + B/(x-2) + C/(x-2)^2

Why do you have to repeat (x-2), instead of just putting B/(x-2)^2?

I know how to solve these types of problems, but I don't understand it conceptually.

Answer 1

Suppose you had to add

1/(x-1) + 1/(x-2) + 1/(x-2)^2

your common denominator would be (x-1)(x-2)^2

in the same way if you had

1/(x-1) + 1/(x-2)^2
you would still have that same common denominator of (x-1)(x-2)^2

So when you are trying to reverse the process, how do you know which way it was?
To allow for all possibilities we use the first version.
In the first case, if the 2nd term is missing, that will show up as B=0

Answer 2

You could solve it as you put it:

1/(x-1)(x-2)^2 = A/(x-1) + B/(x-2) + C/(x-2)^2

or as
1/(x-1)(x-2)^2 = A/(x-1)+(Bx+C)/(x-2)^2

otherwise you'd be missing one parameter of type B/(x-2) in the most general form.

Answer 3

Wow that makes sense. Although, how come the C/(x-2)^2 doesn't follow the rule where the numerator must be one less power than the denominator?

Shouldn't it be Cx+D/(x-2)^2?

Answer 4

When you have B/(x-2), it takes care of the term Bx/(x-2)^2, so you use either one of the two, preferably the former.