## Subtracting and simplifying rational expressions

Complex numbers, rational functions, logarithms, sequences and series, matrix operations, etc.
l-train29
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### Subtracting and simplifying rational expressions

And please show work so I can understand!

Write (x/((x^2)-1)) - ((x+1)/(x-1)) as a single fraction in simplest form.

nona.m.nona
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### Re: Subtracting and simplifying rational expressions

And please show work so I can understand!
For many examples with work shown, please review this lesson on adding and subtracting rational expressions.
Write (x/((x^2)-1)) - ((x+1)/(x-1)) as a single fraction in simplest form.
A good first step might be to factor the first denominator. Then find the least common denominator of the two rational expressions and convert the second rational expression to that denominator.

If you experience difficulties in following these instructions, kindly please reply, showing your efforts (even if you surmise those to be incorrect). Thank you.

l-train29
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### Re: Subtracting and simplifying rational expressions

Hi,

I've been trying to solve this for so long. I had gotten past the first steps, It's the end that seems to be tripping me up. I'l show you what I have so far.

1. I factorised the first denominator to (x+1)(x-1), making the equation (x/((x+1)(x-1)) - ((x+1)/(x-1))

2. The lowest common denominator is therefore (x+1)(x-1). I multiply the second rational expression by (x+1). The equation is now (x/((x+1)(x-1)) - ((x+1)(x+1)/(x-1)(x+1))

3. Simplify to x-((x+1)^2)/(x+1)(x-1)

4. Simplify to x-((x^2)+(2x)+1)/(x+1)(x-1)

5. Simplify to (-x^2)-(x)-1/(x+1)(x-1)

From here I'm not sure what I should do, but I think it could be simpler. That or the above calculations were incorrect.
I want to cancel out the (x-1) but I can't because the numerator isn't multiplying, it's subtracting.

Any advice/tips would be greatly appreciated.

nona.m.nona
Posts: 288
Joined: Sun Dec 14, 2008 11:07 pm
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### Re: Subtracting and simplifying rational expressions

5. Simplify to (-x^2)-(x)-1/(x+1)(x-1)

From here I'm not sure what I should do, but I think it could be simpler.
As the lesson mentioned, the "simplified" form of many rational expressions is often not, in appearance, much "simpler" than that with which one began. This could be one of those instances.

It should be noted, however, that most instructors and texts prefer to clear leading negatives. In your case, this would mean something along the lines of the following:

$\frac{-x^2\, -\, x\, -\, 1}{(x\, +\, 1)(x\, -\, 1)}\, =\, \frac{-1(x^2\, +\, x\, +\, 1)}{(x\, +\, 1)(-1)(1\, -\, x)}\, =\, \frac{x^2\, +\, x\, +\, 1}{(x\, +\, 1)(1\, -\, x)}$

The numerator is not factorable over the integers, so further simplification is unlikely.