My Math Forum Using LCD to Add and Subtract Rational Expressions
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 October 18th, 2008, 01:52 PM #1 Newbie   Joined: Aug 2008 Posts: 16 Thanks: 0 Using LCD to Add and Subtract Rational Expressions My math book has a guide on how to add and subtract two rational expressions with different denominators. The question is: $\frac{3}{(x-1)(x+1)} + \frac{x}{(x+1)^2}$ I know the LCD's are $(x-1)$ and $(x+1)^2$ the book then says to multiply the numerator and denominator of $\frac{3}{(x-1)(x+1)}$ by $(x+1)$ WHY do I multiply the numerator by $(x+1)$ instead of $(x+1)^2$ if $(x+1)^2$ is supposed to be the LCD? so both expressions end up looking like: $\frac{3(x+1)}{(x-1)(x+1)^2} + \frac{x(x-1)}{(x+1)(x+1)^2}$ Thanks!
 October 18th, 2008, 10:58 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,978 Thanks: 2229 $\frac{3}{(x-1)(x+1)} + \frac{x}{(x+1)^2}$ The LCM of the denominators is $(x-1)(x+1)^2.$ To change the denominator of each fraction to the LCM, you have to multiply the numerator and denominator of the first fraction by $(x+1)$ and of the second fraction by $(x-1).$ Can you see why now?
 October 19th, 2008, 01:33 PM #3 Newbie   Joined: Aug 2008 Posts: 16 Thanks: 0 Re: Using LCD to Add and Subtract Rational Expressions I see. So you don't have to multiply by the LCD to GET the denominators to be the same. And the factors you used to get them to be the same equal also be used in the numerators, correct? Thanks!
 October 19th, 2008, 03:33 PM #4 Global Moderator   Joined: Dec 2006 Posts: 20,978 Thanks: 2229 Correct. I hope you can see how the factors are obtained.
October 20th, 2008, 07:00 AM   #5
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Re: Using LCD to Add and Subtract Rational Expressions

Quote:
 Originally Posted by ruit I see. So you don't have to multiply by the LCD to GET the denominators to be the same. And the factors you used to get them to be the same equal also be used in the numerators, correct?
The working principle is exactly the same as with numbers in arithmetic. It is based upon equivalent fractions. You can multiply numerator and denominator by any value. You simply choose that LEAST value that will make the denominators the same for the purpose of adding like things. Fractions are alike when their denominators are alike, or can be made to be alike [as is being done here.] The LCD is (x-1)(x+1)^2, so you'd multiply the first fraction top and bottom by (x+1), and the second top and bottom by (x - 1).

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