Welcome to Georgia Highlands College Math 97 and Math 99 video tutorials.
In this video segment we’ll answer the question, how do you find the LCD or the
least common denominator of rational expressions.
So the first step in this process is to factor each denominator completely,
because we know that least common denominator we’re looking for factors that
multiply together to make a bigger number. So if you're looking at factors, you
need to have your denominators factored.
You list the factors of the first denominator. Add to this list in step two, so
this list of this first denominator. Add to that list any factors from the
second denominator that doesn’t show up in the first list.
Then you're going to form the product of all of those factors that you’ve
gathered together and that product is the LCD.
Let's take a look at a couple of examples.
All right, so this first example I have we’re going to try to find the LCD
between 3 over 10X squared and 7 over 15X, and then we’re looking for an LCD; we
really only need to be focusing on the denominators here. So we looking for the
LCD between 10X squared and 15X.
So the first step is to factor each of these denominators completely.
So 10X squared is the same thing as 2 times 5 times X times X, that’s its’ prime
factorization there.
And for 15X, the prime factorization is 3 times 5 times X.
All right, so now that we have both of these denominators factored, our second
step is to list the factors of the first denominator.
So our first factors are 2, 5, X, and X.
And the third step in the process is to add to this list, and we want to add any
factors of the second denominator that don't show up in the first. So if I look
at my list of factors in the second denominator, I’ve already got an X listed so
I don't need that and I’ve already got a 5 listed. However, I don't have a 3. So
I need to add it to my list from step two there.
And then the final step in this process is to actually multiply these together
to get the LCD.
So for 10X squared and 15X, the LCD is 2 times 5 times X times X times 3 and you
can leave it in its? factored form or you can multiply it all together and say
that the LCD is 30X squared.
All right, moving on to the example on the right-hand side of the screen, this
one’s a little more complex. We have 7X squared plus 28X. Once again, we’re only
focusing on the denominators, trying to find the least common denominator.
So to factor 7X squared plus 28X we have a greatest common factor of 7X and when
I divide both of those terms by 7X, I'm left with X plus 4 and my next
denominator, X squared plus 8X plus16, well this is a trinomial and my leading
coefficient is positive 1. So I just need to find the factors of 16 that
multiply to make 8, and that is 4 and 4.
So this factors as X plus 4 times X plus 4. All right, so our first factors we
have 7X, X plus 4. Now we move to our second factored denominator and I've got o
1X plus 4, so I don't need that one but I've got this other when hanging out, so
I do need to add that second one to my LCD, or to my list that will form the
LCD.
So the LCD for these denominators is 7 times X times X plus 4 times X plus 4.
You can leave it in that form or you could also call this 7X times X plus 4
squared.
I hope that this has been helpful for you in understanding how to find the least
common denominator of rational expressions. Now you'll probably never use this
process standing alone by itself. What you’ll be using it for is to help you add
and subtract rational expressions with unlike denominators. This is the process
that you go through to find that common denominator which is the least common
denominator that you’re looking for.
So if you have any questions about just the process of finding the least common
denominator, please make sure that you ask your Highlands instructor.
Thank you.
