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(x^2 - 9) / (x - 3)
why cant i just remove the x from (x-3) and remove the square from (x^2 - 9) and then take (x-9) / (-3) and then i get x+3 left. my teacher say this is wrong but it works? i know they want me to write (x^2 - 9) as (x-3)(x+3) instead but i get the same answer from both ways. why is my first way wrong?

because it wont always work and u will complicate shit for urself unnecessarily in the future
just do what he wants you to do

you can't. everything between parenthesis is actually 1 * ( x - 3 ) for example
if you remove it you're applying a ( x - 3 ) / x, which would result 1 - 3/x. that's what you need to keep in mind.

+1

you won't get same answer.
(x²-9)/(x-3)= x+3
The way you're trying to do would result in:
(x-9)/(-3) which is equal to: -x/3 + 3 (which is incorrect, since x-3 is divided by everything on the upper part).
You could do that if it was multiplying, like this: x²*9/x*-3 which would result in: -3x.

think of it like this
if you are calculating (x^2 - 9) / x
then you can write it like
(x^2)/x)-(9/x) which simplifies to x-(9/x)
but, if you cancel the x, then you are left with
x - 9
i hope you can see that x-(9/x) =/= x - 9

When you said 'remove' I think you're trying to divide the top and bottom by x
If you actually do that you will have to divide *everything* by x so it will look like:
x - 9/x
--------
1 - 3/x
This is why you need to do the factorising like your teacher said, it won't work otherwise
GL

first is wrong because if x is after ( the answers most of the time equals 2,3 or 8,4

if you "remove" x from the top and bottom you will have
(x-9/x)/(1-3/x) that is what happens when you divide the whole equation by x.
Factoring is the answer here which is what the suggested answer is. The way you did it has no actual mathematical grounds behind it is the problem. Nothing about it is how algebra works that is why your teacher said it cant be done this way. If you had a slightly different question it would not work for you and with math its more about how you got the right answer and not just about getting it.

That is an excellent question. Once you know you can't do it it's easy to just say "you can't", but it's not quite obvious why. It comes down to how you operate fractions.
Remember the pizzas(or pies or whatever). Let's say you have one quarter pizza and one half pizza. You need to divide slices so that each piece is the same size, and then you can count how many slices of that size you have. You can say:
1/4+1/2=1/4+2/4=3/4
Likewise you can say:
( x^2 - 9) / ( x - 3) = x^2 / ( x - 3) - 9 / ( x - 3)
That's completely legit (although not useful in this case). But if we try doing what you did we could for example do:
1 / 2 = ( 1 + 1) / ( 2 + 2)
This is completely legit, but if we now divide the first 1 with the first 2, we get:
( 1/2 + 1) / 2 = 3/4
You clearly see this doesn't work. But after all this I have just showed you that you can't do this rather than give you an explanation why. And that's why I said your question is excellent. I have studied mathematics quite a bit but I wasn't off the top of my head able to give you a competent explanation. If you want I can try to come up with a simple enough explanation, but rest assured knowing that asking questions like this is the key to understanding mathematics, or anything at all.

I understand. But the way i did it works in this case, is it really just a coincidence?

3 as a solution won't work on that question, You need to find that first.