When can you cancel out terms?

I've noticed that I'm not 100% on when I can cancel terms. Say we got $\displaystyle \frac{x^2+3x-2}{x^2-3x-2}$ can I cancel out all the terms and get 1? What about $\displaystyle \frac{x^3+3x-2}{x^2+3x}$ can I cancel out everything except -2? What if -2 is in the denominator?

Please provide as detailed descriptions as you can. I need to nail this. Thanks.

Re: When can you cancel out terms?

Hey Paze.

The basic idea is that you cancel out terms when you have a common factor on both the numerator and the denominator. So as an example given (x^2 + 2x + 1)/(x+1), we get (x^2 + 2x + 1)/(x+1) = (x+1)(x+1)/(x+1) = (x+1).

Note that you can't do this if the factor is zero, so in the above case if x = -1, then you could not do the factorization (but the expression couldn't be evaluated anyway in that case).

Re: When can you cancel out terms?

If you are adding or subtracting variables you can't cancel them out but if they are in brackets say (3x-2y) and you are dividing by (3x-2y) then they do cancel each other out.

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Re: When can you cancel out terms?

Quote:

Originally Posted by

**chiro** Hey Paze.

The basic idea is that you cancel out terms when you have a common factor on both the numerator and the denominator. So as an example given (x^2 + 2x + 1)/(x+1), we get (x^2 + 2x + 1)/(x+1) = (x+1)(x+1)/(x+1) = (x+1).

Note that you can't do this if the factor is zero, so in the above case if x = -1, then you could not do the factorization (but the expression couldn't be evaluated anyway in that case).

I must be misunderstanding you, see the attached image. With one term in the denominator and numerator, canceling that term does not yield me the correct result.

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Re: When can you cancel out terms?

Ohh I see what you're doing and the answer is no.

Basically (A+B+C)/A = (A/A) + (B/A) + (C/A) = 1 + B/A + C/A using normal distributive laws.

Re: When can you cancel out terms?

Quote:

Originally Posted by

**chiro** Ohh I see what you're doing and the answer is no.

Basically (A+B+C)/A = (A/A) + (B/A) + (C/A) = 1 + B/A + C/A using normal distributive laws.

Yes.. Ooh. I see what you were doing in your example. You were factoring the numerator. I'm wondering when you don't factor the numerator. Like with derivatives when we are at the last step of differentiating some function and we say: $\displaystyle \frac{3h+2h^2}{h}$ this becomes $\displaystyle 3+2h$ and I just can't see when and where you can cancel out terms like that. It seems to me that SOMETIMES you can and SOMETIMES you can't. I can't figure the difference :( I'm guessing that you can only cancel like that when EVERY term in the numerator has the same term as in the denominator?

Re: When can you cancel out terms?

For that example we factor out 3h + 2h^2 = h*(3 + 2h) so we get h(3 + 2h)/h = (h/h)*(3+2h) = 3+2h.

Re: When can you cancel out terms?

Quote:

Originally Posted by

**chiro** For that example we factor out 3h + 2h^2 = h*(3 + 2h) so we get h(3 + 2h)/h = (h/h)*(3+2h) = 3+2h.

Oooh, hold on..So every time I can factor something out, I can cancel?

So for $\displaystyle \frac{3x+2x+1}{x}$ I cannot cancel any terms because I can not factor x out. But with $\displaystyle \frac{3x+2x+x}{x}$ I can say $\displaystyle \frac{x(3+2+1)}{x}$ and thus cancel the x out, correct? :o

Re: When can you cancel out terms?

Yes thats spot on, with the exception that you can't cancel out a zero.

So if x is non-zero in your above cancellation then it's ok, but if x = 0 then you can't cancel it out.

Re: When can you cancel out terms?

Quote:

Originally Posted by

**chiro** Yes thats spot on, with the exception that you can't cancel out a zero.

So if x is non-zero in your above cancellation then it's ok, but if x = 0 then you can't cancel it out.

Thanks a lot! That clears it up. Can't believe I haven't gotten this 100% before.