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 April 2nd, 2018, 02:51 PM #1 Newbie   Joined: Mar 2018 From: Canada Posts: 14 Thanks: 0 Least common denominator Hi everyone. The following is giving me some trouble. (x^2 + 2x)/(x+2) + (x)/(x-1) I’m supposed to find the LCD. Here is what is giving me trouble. Do I have to first factor the numerators and then find LCD? ( that would give me (x-1) as LCD) If I don’t factor the numerators first, I believe I would get (x-1).(x+2) as my answer? So my question is? Are both ways of doing it correct? Or should I actually always factor numerators first? Can this expression have 2 LCD? Or it should always have just one ? Thank you very much! Last edited by Brunob; April 2nd, 2018 at 02:54 PM.
 April 2nd, 2018, 05:02 PM #2 Senior Member   Joined: May 2016 From: USA Posts: 1,310 Thanks: 551 $\dfrac{x^2 + 2x}{x + 2} + \dfrac{x}{x - 1} = \dfrac{(x^2 + 2x)(x - 1)}{(x + 2)(x - 1)} + \dfrac{(x + 2)x}{(x + 2)(x - 1)} =$ $\dfrac{x^3 - x^2 + 2x^2 - 2x + x^2 + 2x}{(x + 2)(x - 1)} = \dfrac{x^2(x + 2)}{(x + 2)(x - 1)} = \dfrac{x^2}{x - 1}.$ ALTERNATIVELY. $\dfrac{x^2 + 2x}{x + 2} + \dfrac{x}{x - 1} = \dfrac{x(x + 2)}{x + 2} + \dfrac{x}{x - 1} =$ $x + \dfrac{x}{x - 1} = \dfrac{x(x - 1)}{x - 1} + \dfrac{x}{x - 1} = \dfrac{x^2 - x + x}{x - 1} = \dfrac{x^2}{x - 1}.$ It makes absolutely no difference to the result which way you do it. They are equivalent. It is usually easier to simplify the fractions first if possible.
April 2nd, 2018, 05:48 PM   #3
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Quote:
 Originally Posted by Brunob (x^2 + 2x)/(x+2) + (x)/(x-1) If I don’t factor the numerators first, I believe I would get (x-1).(x+2) as my answer?
That is not correct (as Jeff told you) BUT you could
have known that by assigning a value to x, like x=5; then:

(x^2 + 2x)/(x+2) + (x)/(x-1) = 6.25

(x-1).(x+2) = 28

Doing this during simplification tells you if
you're on the right path or not...get it, tabarnak?!

April 2nd, 2018, 07:22 PM   #4
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 Originally Posted by Denis That is not correct (as Jeff told you) BUT you could have known that by assigning a value to x, like x=5; then: (x^2 + 2x)/(x+2) + (x)/(x-1) = 6.25 (x-1).(x+2) = 28 Doing this during simplification tells you if you're on the right path or not...get it, tabarnak?!
Denis, You are mistaken. The OP just asked for the LCD and claimed the answer (x-1).(x+2) is correct. As a result your substitution was not necessary. So what do you have to say about them apples?

April 2nd, 2018, 07:26 PM   #5
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Quote:
 Originally Posted by Brunob Hi everyone. The following is giving me some trouble. (x^2 + 2x)/(x+2) + (x)/(x-1) I’m supposed to find the LCD. Here is what is giving me trouble. Do I have to first factor the numerators and then find LCD? ( that would give me (x-1) as LCD) If I don’t factor the numerators first, I believe I would get (x-1).(x+2) as my answer? So my question is? Are both ways of doing it correct? Or should I actually always factor numerators first? Can this expression have 2 LCD? Or it should always have just one ? Thank you very much!
I'll let you decide. You want the LEAST common Denominator and came up with two Common Denominators, namely (x-1).(x+2) and (x-1). And the least one is....?

Consider this problem: 2(x-3)/(x-3) + 3(x+7)/(x+7). How would you do this one? I would just do 2+3=5 and be done.

Last edited by Jomo; April 2nd, 2018 at 07:29 PM.

April 3rd, 2018, 04:10 AM   #6
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 Originally Posted by Jomo Denis, You are mistaken. The OP just asked for the LCD and claimed the answer (x-1).(x+2) is correct. As a result your substitution was not necessary. So what do you have to say about them apples?
Whoops...that'll teach me to read a problem in full!!
I'll go stand in the corner for (x-1)(x+2) minutes...
Thanks for the reprimand, Oh Jomo

April 3rd, 2018, 04:11 PM   #7
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 Originally Posted by Denis Thanks for the reprimand, Oh Jomo
You never have to worry about me missing an opportunity to reprimand Sir Denis.

April 3rd, 2018, 06:00 PM   #8
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 Originally Posted by Jomo You never have to worry about me missing an opportunity to reprimand Sir Denis.
And there are so many opportunities!

-Dan

April 3rd, 2018, 07:36 PM   #9
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Quote:
 Originally Posted by Jomo two Common Denominators, namely (x-1).(x+2) and (x-1). And the least one is....?
Well if x = - 1, then (x - 1)(x + 2) = - 2 * 1 = - 2. But (x - 1) = - 2. Which is lesser? Under those circumstances [2(x - 1)(x + 2)]/2 would be less than either, so they certainly aren't least.

It is, I think, a very stupid question. I suspect that what was meant was the simplest common denominator. In any case, I carefully avoided answering it.

Last edited by JeffM1; April 3rd, 2018 at 07:39 PM.

 April 4th, 2018, 06:08 PM #10 Newbie   Joined: Mar 2018 From: Canada Posts: 14 Thanks: 0 Thanks everyone!

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