When find LCD for a rational expressions.

In summary, the conversation was about finding the Lowest Common Denominator (LCD) for three rational expressions with denominators x^2 - 4 and x - 2. The expert explains that the LCD must be the smallest common multiple of the two denominators, which is x^2 - 4. This is found by multiplying the two denominators and canceling out any common factors. The expert also clarifies that this method only works if the two polynomials are prime with each other.
  • #1
DLxX
58
0
Ok I have to find the Lowest Common Denominator for 3 ration expressions.
I don't think the numerators are important so Ill just leave them out. The denominators are

x^2 - 4 and x -2 I got the LCD as x-2. correct?

When finding the LCD in expressions like this you just have to factor and pick the term that they both have in common right?
 
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  • #2
Nope,u have to pick just that:the lowest common denominator.Think of the numbers:what is the lowest common multiple of 6 and 3...?Is it 3??

Daniel.
 
  • #3
Well the first one comes out to (x-2) and (x+2) while the 2nd one just stays as (x-2). So isn't the only CD they have (x-2)?
 
  • #4
No,because the common denominator has to be a common multiple of the "old" denominators...That's why i gave the example with the numbers...Instead of #,you have polynimials...

Daniel.
 
  • #5
Would you mind just giving me the answer for the one I mentioned? I have 3 more problems just like it and right now I am seriously not getting it.
 
  • #6
It would really matter for you to know/understand why the answers is the one that it is,namely [tex] x^{2}-4 [/tex].

Daniel.
 
  • #7
Ok I think I get it now. You take x-2 from the first one and x-2 from the 2nd one and multiply them right? But wouldn't that end up x^2 -4x + 4? Why do I take the x+2 from the first one?

EDIT: I take all the DIFFERENT factors right? So that they each get a part?
 
  • #8
Nope,it has to be the smallest common denominator,as i said the smalles common MULTIPLE OF THE 2 DENOMINATOS.The smallest one for [itex] x^{2}-4 [/itex] is [itex] x^{2}-4 [/itex] and that's that...

That simple multiplication between the denominators would be valid if the 2 polynomials would be prime one wrt another,which is not the case in here...

Daniel.
 
  • #9
So I can essentially just multiply the 2 denominators of any problem like this and still get the correct answer? Or factor out the 2 denominators and multiply all the factors with each other, crossing out all the factors that were in both?
 
  • #10
Yes,you finally got it...That "crossing out" doesn't mean eliminating,just "counting" only once in the product,okay?
In your case,you'd have to count "x-2" only once.

Daniel.
 

1. What is LCD in rational expressions?

The LCD (lowest common denominator) in rational expressions is the smallest number that can be divided evenly by all the denominators in the expressions. It is used to simplify fractions and make it easier to add or subtract rational expressions.

2. How do I find the LCD for rational expressions?

To find the LCD for rational expressions, first factor each denominator completely. Then, identify the common factors among all the denominators. The LCD is the product of these common factors.

3. Why is finding the LCD important in rational expressions?

Finding the LCD allows us to simplify fractions and make it easier to add or subtract rational expressions. It also helps us to avoid working with large, complex fractions.

4. Can the LCD be different for each rational expression?

Yes, the LCD can be different for each rational expression. It depends on the denominators of the expressions and their common factors.

5. Are there any shortcuts for finding the LCD of rational expressions?

One shortcut for finding the LCD is to use the prime factorization method. This involves finding the prime factors of each denominator and then multiplying the highest powers of each prime factor. Another shortcut is to use the "cross-multiplying" method, where you multiply each denominator by the other denominators' factors that it is missing. However, it is important to note that these methods may not always work and it is best to factor the denominators completely to find the LCD.

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