i don't get this.. please explain [math]?

Question

x^-3 - x
----------------
x^-2 - 1

"Take x common from the numerator :
x(x^2 - 1)"

i don't get it.. x^-2 -1 has no nothing in common :D

i have the answer from the book..

1+x^2
-----------
x

i tried solving it.. but i get something else..

i hate math >:P

Answer 1

They worded it really weird but I'll try to explain as best I can.

The numerator (top half of the equation) contains two different terms (terms are seperated by either addition or subtrction symbols):
The first is x^-3 and the second term is -x.
Now, when they say that both terms have an x in common, it means that both terms have an x in them. This is important because:

Let's rewrite the two terms:
First term: x^-3 is equal to 1 / x^3 is equal to 1 / xxx right? There are 3 x's there.
Second term: -x There is one x there.

So, we can factor out what both terms contain - both terms contain a single x. So, when we take out that x...

First term: becomes x^-2 or 1 / xx
Second term: becomes -1 (the number that was in front of the x before we got rid of it).

So x^-3 - x becomes x(x^-2 - 1) which is just another way of rewriting the problem because when you multiply the x back into the parenthesis you get the same thing you started with! So why did we do all that? Well...when you have the same number above and below the parenthesis, you can cancel it out. we now have (x^-2 - 1) both above and below...when it divides out both terms become 1...now look how simple the problem has become! Much better...

Hopefully that answers you question for you. Write me if you still have a problem with this problem! :o)

Answer 2

You have (x^-3 - x) / (x^-2 - 1)
= x * (x^-4 - 1) / (x^-2 - 1)
We can factorize
x^-4 - 1 = (x^-2 - 1)*(x^-2 + 1)
ie difference of two squares
So the expression becomes
x * (x^-2 - 1)*(x^-2 + 1) / (x^-2 - 1)
= x * (x^-2 + 1)
= x * (1/x^2 + 1)
= x * (1 + x^2) / x^2
= (1 + x^2) / x

Answer 3

= (1/x³ - x) / (1/x² - 1)

= (1 - x^4) / (x - x ³)

= [ (1 - x²)(1 + x²) ] / [ x (1 - x²) ]

= (1 + x²) / x

= 1/x + x

Answer 4

Take x common from the numerator :
x(x^2 - 1)
Now cancel from the denominator to get the answer as x.

Hope this helps.

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Answer 5

x^-3-x/x^-2-1

=x(x^-4-1)
--------------
x^-2 -1
=x[(x^-2-1)(x^-2+1)]
-------------------------
x^-2-1
=x(x^-2+1)

Answer 6

x^-3 = 1/x^3

x^-2 = 1/x^2