WEBVTT mathematics/algebra-2/eaton
00:00:00.000 --> 00:00:02.300
Welcome to Educator.com.
00:00:02.300 --> 00:00:06.300
Today, we are going to be working on adding and subtracting rational expressions.
00:00:06.300 --> 00:00:16.800
And recall that rational expressions are algebraic fractions; so this is more complicated than multiplying or dividing rational expressions.
00:00:16.800 --> 00:00:27.600
But we are just going to apply the same techniques that we used to add or subtract fractions involving numbers to add or subtract rational expressions.
00:00:27.600 --> 00:00:31.900
Let's start out by reviewing some concepts from working with fractions.
00:00:31.900 --> 00:00:37.900
Recall the idea of the least common multiple, because we are going to use that to get a common denominator.
00:00:37.900 --> 00:00:46.400
As you know, if you have a common denominator with your fractions (such as 2/4 + 1/4), it is very easy to add.
00:00:46.400 --> 00:00:52.400
You just add the numerators (2 + 1 is 3) and use the common denominator.
00:00:52.400 --> 00:00:58.500
It is the same with rational expressions: if you already have a common denominator, add the numerators
00:00:58.500 --> 00:01:09.200
and use the common denominator as the denominator of the rational expression that you end up with from adding or subtracting.
00:01:09.200 --> 00:01:13.200
However, just like with fractions, when you don't have a common denominator,
00:01:13.200 --> 00:01:21.300
you have to find a common denominator and convert to equivalent fractions before you can add or subtract.
00:01:21.300 --> 00:01:31.200
Let's talk about an example using numbers: if I want to add 1/10 and 2/45, you might just look at this,
00:01:31.200 --> 00:01:36.500
know what the common denominator is, quickly convert it, possibly even in your head, and add it.
00:01:36.500 --> 00:01:44.900
But I am going to bring out each step of that consciously, so we can use that to work with adding and subtracting rational expressions.
00:01:44.900 --> 00:01:53.000
Let's look at the denominators: let's find the least common multiple of these two numbers, 10 and 45.
00:01:53.000 --> 00:01:57.700
In order to find a common multiple, you need to factor.
00:01:57.700 --> 00:02:03.600
And whether we are working with a number, or if the denominator is a polynomial (as in a rational expression),
00:02:03.600 --> 00:02:06.700
the idea is still that you have to factor.
00:02:06.700 --> 00:02:17.100
So, let's factor this out: and I get 2 times 5 here; for 45, I might say it's 9 times 5, but I can go farther with my factoring.
00:02:17.100 --> 00:02:23.900
9 factors into 3 times 3; I could also rewrite this as 3² times 5.
00:02:23.900 --> 00:02:32.500
So, I have 2 times 5 for 10, and then I have 3² times 5 as my prime factorization for 45.
00:02:32.500 --> 00:02:41.200
The least common multiple is going to be the product of the unique factors that I see here,
00:02:41.200 --> 00:02:50.200
to the highest power that they appear for any one number (or, when we are talking about polynomials, for any one polynomial).
00:02:50.200 --> 00:02:58.800
So, the unique factors: 2 is a factor, and I am going to take the product of that and the other factors I see.
00:02:58.800 --> 00:03:08.000
5 is here once only; and it is here once only; so the greatest number of times that 5 appears in any one set of factors is once.
00:03:08.000 --> 00:03:12.300
So, I am not going to write it twice: I am just going to write it once.
00:03:12.300 --> 00:03:16.800
I am done with this one; here I still have another factor that isn't accounted for; I have a 3.
00:03:16.800 --> 00:03:20.700
But it appears twice, so I am going to put 3².
00:03:20.700 --> 00:03:26.100
Now, if I had 5 twice up here, then I would have written 5²; I only had it once.
00:03:26.100 --> 00:03:35.000
All right, this equals...2 times 5 is 10, times 3²...so 10 times 9 is 90.
00:03:35.000 --> 00:03:41.300
The least common multiple of these two is 90; and I am going to end up using that as my common denominator.
00:03:41.300 --> 00:03:48.200
But as you know, I can't just change this to 1/90 and add that to 2/90, because those fractions, then, won't be the same.
00:03:48.200 --> 00:03:54.400
So, we are going to talk about, once you get the common denominator that you are going to use, what to do with the rest of the fraction.
00:03:54.400 --> 00:04:01.200
And we will talk about that in a minute; but first, let's apply this concept to finding the least common multiple of polynomials.
00:04:01.200 --> 00:04:10.100
If I was working with something like 1 over x² plus 7x, plus 14, and I wanted to add that
00:04:10.100 --> 00:04:19.900
to 1 over 2x² - 2x - 12, then I need to find the common multiple of these two denominators.
00:04:19.900 --> 00:04:31.900
So, to find the LCM, I am going to factor out both of these denominators, just the way I did up here with the numbers.
00:04:31.900 --> 00:04:42.300
OK, so this is going to give me x, and then x; I have all plus signs, so I know that these are going to be positive.
00:04:42.300 --> 00:04:57.700
Factors of 14...actually, let's make this a 10...let's go ahead and make that a 10 for much easier factorization.
00:04:57.700 --> 00:05:13.000
This is going to be factors of 10: 1 and 10, and 2 and 5; and I need them to add up to 7, and 2 + 5 = 7.
00:05:13.000 --> 00:05:19.500
OK, so this is factored out as far as I can go.
00:05:19.500 --> 00:05:25.200
Now, down here, I am aware that I have a common factor of 2;
00:05:25.200 --> 00:05:40.900
so I am going to pull that out to get 2(x² - x...and that is going to give me - 6).
00:05:40.900 --> 00:05:51.100
Now, work on factoring this: 6 has factors that are 1 and 6, and 2 and 3.
00:05:51.100 --> 00:05:58.900
I have an x here; I have an x here; and since this is a negative, I have to have a positive and a negative.
00:05:58.900 --> 00:06:03.800
And I want them to add up to -1, so I need to find factors that are close together.
00:06:03.800 --> 00:06:08.400
And I need to make the larger factor negative, so that this will add up to -1.
00:06:08.400 --> 00:06:17.100
Let's try 2 - 3; that equals -1; so I am going to make the 2 positive and the 3 negative.
00:06:17.100 --> 00:06:26.100
OK, now I factored out both denominators; and this is going to then give me my least common multiple.
00:06:26.100 --> 00:06:29.000
I am going to look for each factor that appears.
00:06:29.000 --> 00:06:33.800
And I see an x + 2; it appears here once, and it only appears here once.
00:06:33.800 --> 00:06:41.500
So, the highest power to which it appears is just 1, so it is x + 2.
00:06:41.500 --> 00:06:51.800
I also have an x + 5; it appears once, and I am going to put it once.
00:06:51.800 --> 00:06:56.500
This is done; I accounted for this; I also need an x - 3.
00:06:56.500 --> 00:07:07.200
Therefore, the least common multiple of these denominators is going to be (x + 2)(x + 5)(x - 3)...
00:07:07.200 --> 00:07:18.000
and let's not forget about the 2; the 2 needs to come along as well, so 2(x + 2)(x + 5)(x - 3).
00:07:18.000 --> 00:07:27.800
If I had a situation where I factored something out, and I got (x + 3)(x + 3)(x - 1), and then I had down here
00:07:27.800 --> 00:07:41.200
(x + 3)(x - 5), then the LCM would be...I have an (x + 3), but it appears twice in this polynomial, so it would be (x + 3)².
00:07:41.200 --> 00:07:48.600
And then, I would have my x - 1 and x - 5.
00:07:48.600 --> 00:07:55.800
So, the least common multiple will allow you to find a common denominator when you are adding or subtracting rational expressions.
00:07:55.800 --> 00:07:58.800
To add or subtract, first find a common denominator.
00:07:58.800 --> 00:08:03.900
The LCM of the denominators is the smallest value that we can use as a common denominator.
00:08:03.900 --> 00:08:07.200
And we are going to refer to that as the least common denominator.
00:08:07.200 --> 00:08:12.400
The least common denominator is just the LCM of the two denominators.
00:08:12.400 --> 00:08:17.300
OK, so let's go back to our example with numbers.
00:08:17.300 --> 00:08:25.600
And I had 1/10, and I asked you to add 2/45.
00:08:25.600 --> 00:08:36.600
We factored out 10 to get 2 times 5; we factored out 45 to get 3² times 5.
00:08:36.600 --> 00:08:47.900
The LCM...a 2 appears up here; 5 appears, at most, once; and 3 appears twice, so it is 3².
00:08:47.900 --> 00:08:58.400
This equals 90; now, I am going to use this 90 as the least common denominator; so, the least common denominator for these two fractions is 90.
00:08:58.400 --> 00:09:04.300
But I can't simply say, "OK, I am going to make this 1/90 and 2/90"; those won't be the same fractions.
00:09:04.300 --> 00:09:07.900
I need to convert these to equivalent fractions, but with my new denominator.
00:09:07.900 --> 00:09:12.100
Again, this is something you just know how to do; but we need to bring it out and look at each step.
00:09:12.100 --> 00:09:19.500
So, let's talk about converting 1/10: if I rewrite this as 1/(2 times 5), it is factorization.
00:09:19.500 --> 00:09:25.800
That allows me to look at the LCM and determine what I am missing--what factor am I missing?
00:09:25.800 --> 00:09:32.400
I have a 2; I have a 5; but I am missing the 3².
00:09:32.400 --> 00:09:37.200
The thing is that I can't just multiply the denominator alone; what I am allowed to do
00:09:37.200 --> 00:09:43.700
is multiply both the numerator and the denominator by the same number, because this is just 1--
00:09:43.700 --> 00:09:47.300
this cancels out to 1--and I am allowed to multiply by 1.
00:09:47.300 --> 00:10:01.800
So, this is 1 times 9; this is 2 times 5 is 10, times 9 is 90; and of course, that simplifies to 1/10, so I know I have a fraction that still has the same value.
00:10:01.800 --> 00:10:10.800
2/45 can be written as 2 over (3² times 5); what is missing from this denominator?
00:10:10.800 --> 00:10:21.000
Well, I have a 5; and I have a 3²; I look down here--the 2 is missing.
00:10:21.000 --> 00:10:26.900
So, I need to multiply both the numerator and the denominator by 2.
00:10:26.900 --> 00:10:33.500
2 times 2 is 4, over 9 times 5 is 45, times 2...so that is 4/90.
00:10:33.500 --> 00:10:41.400
Now, I can add these two: I have 9/90 + 4/90; that is 13/90.
00:10:41.400 --> 00:10:45.800
We are doing the same thing when we add rational expressions or subtract rational expressions.
00:10:45.800 --> 00:10:51.200
We are going to take the same steps: we are going to find the least common multiple, and that will be our common denominator.
00:10:51.200 --> 00:10:58.500
Then we are going to convert to fractions that are equivalent to the original fractions, but with the new denominator.
00:10:58.500 --> 00:11:03.100
And then, we can add or subtract those once they have the same denominator.
00:11:03.100 --> 00:11:18.500
OK, looking at an example using rational expressions: here is my first rational expression, and I am going to add that to (x + 4)/(2x - 6).
00:11:18.500 --> 00:11:31.800
My first step is to find the LCD: factor out the denominators, and then find the least common multiple to use that as the denominator.
00:11:31.800 --> 00:11:43.500
So, I have my two denominators: this factors into, since it has a negative, (x + something) (x - something).
00:11:43.500 --> 00:11:48.200
The only factors of 3 that I am going to have to work with are 1 and 3.
00:11:48.200 --> 00:11:56.700
Since I want to end up with a negative 2x, I am going to make the larger number negative: 1 - 3 is -2.
00:11:56.700 --> 00:12:07.800
So, I am going to make this a 2: (x + 2) (x - 3), and (x + 1) (x - 3); so that is
00:12:07.800 --> 00:12:18.000
x² - 3x + 1x (so that is -2x), and then the last terms multiply out to -3.
00:12:18.000 --> 00:12:36.200
OK, and this should be 2x - 6, so I am going to look at the common facto that I have, which is 2.
00:12:36.200 --> 00:12:51.500
So, this factors out to 2(x - 3); the LCM is the product of the factors of both of these.
00:12:51.500 --> 00:13:00.100
I have (x + 1), and that only appears once; that is the highest power to which it appears in any of these polynomials.
00:13:00.100 --> 00:13:02.100
So, I am just going to put (x + 1).
00:13:02.100 --> 00:13:09.200
I also have an (x - 3), and that is present once here and once here, so I represent it once.
00:13:09.200 --> 00:13:17.100
Down here, I also have a 2; I am going to pull that out in front; the LCM is 2(x + 1) (x - 3).
00:13:17.100 --> 00:13:25.000
And I am going to use it as the least common denominator.
00:13:25.000 --> 00:13:33.700
We found the common denominator; next we need to find the equivalent fractions, using that common denominator (using the LCD as their denominator).
00:13:33.700 --> 00:13:42.000
Once I have done that, all I have to do is add or subtract the numerators of the fractions, and then simplify.
00:13:42.000 --> 00:13:59.500
Let's continue on with the example I started in the previous slide, which was 1/(x² - 2x - 3) + (x + 4)/(2x - 6).
00:13:59.500 --> 00:14:18.800
This factored out to 1/(x - 3) (x + 1).
00:14:18.800 --> 00:14:29.000
This, you recall, factored into 2(x - 3).
00:14:29.000 --> 00:14:33.900
So, I factored these out; I haven't done anything else with them yet--I have left them the same, but I have factored them out.
00:14:33.900 --> 00:14:42.200
So, I have them factored: the LCD is going to be the product of the unique factors.
00:14:42.200 --> 00:14:52.800
Remember, I had an (x - 3) as part of that; I had an (x + 1), which appears here; I already have my (x - 3) accounted for; and I need a 2.
00:14:52.800 --> 00:15:04.400
So, I am going to rewrite this with a 2 out in front; and I am going to use my least common multiple as my denominator, my LCD.
00:15:04.400 --> 00:15:09.200
2 times (x - 3) times (x + 1)...
00:15:09.200 --> 00:15:12.300
The hard part can be converting to equivalent fractions.
00:15:12.300 --> 00:15:17.400
You just have to be careful and make sure that you multiply the numerator and the denominator by the same thing.
00:15:17.400 --> 00:15:22.100
Otherwise, you will not end up with an equivalent fraction.
00:15:22.100 --> 00:15:34.700
To convert to equivalent fractions is my next step.
00:15:34.700 --> 00:15:39.200
I am going to take this first fraction, and I am going to say, "OK, what is lacking?"
00:15:39.200 --> 00:15:44.100
I want to turn this denominator into this.
00:15:44.100 --> 00:15:49.200
I have my (x - 3); I have my (x + 1); but I am missing a 2.
00:15:49.200 --> 00:16:00.100
Therefore, I am going to multiply both the numerator and the denominator by 2/2, which is just multiplying this fraction by 1.
00:16:00.100 --> 00:16:13.700
The second fraction: to turn this denominator into what I have here, I have to figure out what I am missing.
00:16:13.700 --> 00:16:18.200
I have a 2; I have an (x - 3); I am missing the (x + 1).
00:16:18.200 --> 00:16:24.600
So, I am going to multiply both the numerator and the denominator by (x + 1).
00:16:24.600 --> 00:16:27.900
Again, this is just multiplying by 1.
00:16:27.900 --> 00:16:46.200
This is going to give me...1 times 2 is 2, over (x - 3) (x + 1) (2)...and I am pulling the 2 out in front.
00:16:46.200 --> 00:17:02.400
Here, I am going to add that; and in the numerator, I am going to have (x + 4) (x + 1) for the second fraction, for the numerator.
00:17:02.400 --> 00:17:11.400
In the denominator, I am going to have that same common denominator: 2/(x - 3)(x + 1).
00:17:11.400 --> 00:17:21.600
So, I have just multiplied the numerator and the denominator by whatever was missing from the denominator to form the LCD.
00:17:21.600 --> 00:17:27.500
Now that I have a common denominator, I just add or subtract the numerator of these fractions.
00:17:27.500 --> 00:17:45.500
So, since I am adding, this is going to become 2 + (x + 4)(x + 1), all over my common denominator.
00:17:45.500 --> 00:17:55.800
In the same way, if I am adding 3/4 and 5/4, I just add the numerators and put them over the common denominator.
00:17:55.800 --> 00:18:04.200
The same idea here--I added the numerators and put them over the common denominator.
00:18:04.200 --> 00:18:14.600
Finally, I am going to simplify: I am going to need to multiply this out, and this is going to give me 2 +...let's work up here:
00:18:14.600 --> 00:18:26.100
(x + 4) times (x + 1): using FOIL, this is going to give me x², and then the outer term is going to be + x;
00:18:26.100 --> 00:18:31.200
then inner term is going to be 4x, so that is 5x; and then, the last term is going to be 4.
00:18:31.200 --> 00:18:49.200
So, this is going to give me x² + 5x + 4, all over 2 times (x - 3)(x + 1).
00:18:49.200 --> 00:18:53.600
So, I am working right over here, just to do that multiplication.
00:18:53.600 --> 00:19:01.600
(x - 3) times (x + 1)...and then we are going to have to multiply all of that by 2.
00:19:01.600 --> 00:19:10.800
That is going to give me...actually, we are going to go ahead and leave that in factored form, because that will make it easier to simplify.
00:19:10.800 --> 00:19:17.500
Let's just go ahead and leave that one in factored form.
00:19:17.500 --> 00:19:34.300
So, the top gives me x², and...I have 5x; I leave that alone; and this is 2 + 4, to give me 6, all over this.
00:19:34.300 --> 00:19:38.800
Now, at the top, I went ahead and multiplied this out, so I could add it to this.
00:19:38.800 --> 00:19:45.000
For the denominator, I didn't need to do that, because it is already factored out; and you will see why in a second.
00:19:45.000 --> 00:19:49.600
OK, can I factor this? Well, let's go ahead and see.
00:19:49.600 --> 00:20:01.100
I have x and x, and right here, I have positives; so I am going to make that a + and make that a +.
00:20:01.100 --> 00:20:07.200
6...what are my factors? 1 and 6, 2 and 3.
00:20:07.200 --> 00:20:19.600
So, I am looking for factors of 6 that are going to end up giving me 5; and I can see that 2 + 3 = 5, so this is going to be (x + 3) (x + 2).
00:20:19.600 --> 00:20:27.700
That is x² + 2x + 3x (is 5x) + 6.
00:20:27.700 --> 00:20:31.800
OK, now you can see why I just left the denominator in factored form.
00:20:31.800 --> 00:20:35.500
I look here, and I see if I can simplify--are there any common factors?
00:20:35.500 --> 00:20:39.300
And there are actually not--I don't have any common factors in the numerator or denominator.
00:20:39.300 --> 00:20:42.800
And it is perfectly fine to leave the expression like this.
00:20:42.800 --> 00:20:46.500
You might want to go on and do something else with it, and then it is already factored.
00:20:46.500 --> 00:20:54.500
For something like we had in the numerator, where we ended up with 2 plus all of this, we need to actually combine that.
00:20:54.500 --> 00:20:58.000
So, I needed to multiply this out, add, and then factor.
00:20:58.000 --> 00:21:02.100
This was already factored out; so I left it, and this was my final answer.
00:21:02.100 --> 00:21:07.700
So again, I found the LCD of these two by factoring.
00:21:07.700 --> 00:21:18.800
This was the LCD; I converted the first and the second fractions into equivalent fractions, and then I just added and tried to simplify.
00:21:18.800 --> 00:21:24.000
OK, we have talked before about simplifying complex fractions, but this time
00:21:24.000 --> 00:21:30.600
we are going to talk about complex fractions that also may require you to add or subtract in the numerator or denominator.
00:21:30.600 --> 00:21:36.700
So, first I want to explain the difference between what we did in a previous lesson and what we are going to do now.
00:21:36.700 --> 00:21:53.700
In the previous lesson, we talked about things like this, rational expressions that are complex fractions: (3xy/(x² - 16))/ ((2x + 1)/xy).
00:21:53.700 --> 00:22:10.000
So, the steps for this were to rewrite this as division, because I know that this fraction bar is telling me to take the numerator and divide it by the denominator.
00:22:10.000 --> 00:22:32.000
Once you got to this point, you recognized that this is simply the first rational expression, times the reciprocal of the second.
00:22:32.000 --> 00:22:37.300
And I am not going to do the whole multiplication right now; I just wanted to show you the setup on that.
00:22:37.300 --> 00:22:44.200
So, if I had a complex fraction, such as this, I just took the numerator, set it as divided by the denominator,
00:22:44.200 --> 00:22:52.700
and then converted it to multiplication of the first rational expression times the inverse of the second.
00:22:52.700 --> 00:22:57.300
OK, what we are talking about here is actually different.
00:22:57.300 --> 00:23:04.700
It is more complex, and requires one more step before you can go from here to here.
00:23:04.700 --> 00:23:20.300
Let's say I have something like this: (1/x + 2/y)/(3/x - 1/y).
00:23:20.300 --> 00:23:28.500
Now, not only do I have a complex fraction (I have a situation where I have a fraction in the numerator and a fraction in the denominator),
00:23:28.500 --> 00:23:33.200
but I have a sum and/or a difference (here I have a sum; here I have a difference).
00:23:33.200 --> 00:23:38.800
I have a fraction up here that I am adding to another fraction; I have a fraction down here that I am subtracting from another fraction.
00:23:38.800 --> 00:23:46.000
Up here, I just had one fraction; I could just go ahead and simplify by multiplication; down here, I just had one fraction.
00:23:46.000 --> 00:23:50.300
So, the difference here is: we are talking about simplifying complex fractions
00:23:50.300 --> 00:23:56.900
in which you need to add or subtract in the numerator or the denominator of the complex fraction.
00:23:56.900 --> 00:23:59.500
What you are going to do is handle these separately.
00:23:59.500 --> 00:24:03.600
You are going to handle the numerator; you are going to handle the denominator; and then you are going to put it all together.
00:24:03.600 --> 00:24:11.200
So, to simplify a complex fraction, add or subtract the fractions in the numerator and the denominator separately, and then simplify.
00:24:11.200 --> 00:24:17.200
I am going to start out with the numerator.
00:24:17.200 --> 00:24:24.600
In the numerator, I have 1/x + 2/y; I need to find a common denominator.
00:24:24.600 --> 00:24:37.400
And since all I have in the denominator is an x, and all I have in the denominator over here is a y, then my LCD (or my LCM) is going to be xy.
00:24:37.400 --> 00:24:45.200
Now, I need to convert these to equivalent fractions.
00:24:45.200 --> 00:24:54.700
I see that what I am lacking from this denominator is a y, so I need to multiply this times y/y.
00:24:54.700 --> 00:25:01.300
Now, I have an xy in the denominator, and I multiplied the numerator by the same thing, so that I end up with equivalent fractions.
00:25:01.300 --> 00:25:08.200
Over here, I want to get the LCD of xy; what I am lacking in the denominator is an x.
00:25:08.200 --> 00:25:12.700
So, I am going to multiply both the numerator and the denominator by x.
00:25:12.700 --> 00:25:26.400
This is going to give me y/xy, plus 2x/xy.
00:25:26.400 --> 00:25:28.500
Now, I can add those, because they have the same denominator.
00:25:28.500 --> 00:25:34.500
And it is just going to be (y + 2x)/xy.
00:25:34.500 --> 00:25:46.200
Now, I go back here, and what I have in the numerator now is (y + 2x)/xy.
00:25:46.200 --> 00:25:56.000
I no longer have the sum, where I have two separate fractions: I have one fraction in the numerator.
00:25:56.000 --> 00:26:16.900
OK, denominator: the same thing--the denominator is 3/x - 1/y.
00:26:16.900 --> 00:26:27.700
Again, I have an LCD down here--my LCD is just going to be xy, because this is the only factor here; this is the only factor here.
00:26:27.700 --> 00:26:38.200
I need to convert these to equivalent fractions, so what I am going to do is say, "All right, what am I lacking from this?"
00:26:38.200 --> 00:26:45.700
I am lacking a y in the denominator, so I am going to multiply this times y/y.
00:26:45.700 --> 00:26:52.400
And I am subtracting that from 1/y, and what I am lacking in the common denominator is an x.
00:26:52.400 --> 00:27:09.600
So, I am going to multiply this times x/x; this is going to give me 3y/xy - x/xy.
00:27:09.600 --> 00:27:23.600
I now have a common denominator: this gives me (3y - x)/xy: this is my denominator, (3y - x)/xy.
00:27:23.600 --> 00:27:28.700
That was the hard part: once you get to here, you are working with this situation.
00:27:28.700 --> 00:27:36.700
I could handle this by rewriting this as a division problem: (y + 2x)/xy--I am going to take that;
00:27:36.700 --> 00:27:43.000
it is going to be like this first rational expression; and I am going to say "divided by" this whole thing.
00:27:43.000 --> 00:27:51.700
Once you get to that point, you use our usual method of taking the first rational expression and multiplying it by the inverse of the second.
00:27:51.700 --> 00:27:58.100
This is the difficult step: and the way to really look at this is to handle the numerator and the denominator separately.
00:27:58.100 --> 00:28:04.000
Your goal is to get the numerator to look like this (a single fraction) and the denominator to look like this.
00:28:04.000 --> 00:28:12.700
This is an addition problem with a rational expression; this is an addition problem with a rational expression.
00:28:12.700 --> 00:28:21.900
And once you take care of the numerator and you take care of the denominator, then you can proceed as usual.
00:28:21.900 --> 00:28:31.700
This is a complex problem: it just has a lot of steps, and you just need to take it one at a time and keep track of what you are working with.
00:28:31.700 --> 00:28:38.000
OK, with the examples, we are going to start out just finding the LCM; but this time, it is going to be of three polynomials.
00:28:38.000 --> 00:28:47.800
So, when we find the LCM, no matter if it is 2, 3, 4, or more, we need to factor.
00:28:47.800 --> 00:29:03.600
I am going to factor the first polynomial, the second polynomial, and the third polynomial.
00:29:03.600 --> 00:29:10.200
OK, so this is x; and I have a negative sign here, but I have a positive sign here.
00:29:10.200 --> 00:29:15.200
That clues me into the fact that I have a negative and a negative.
00:29:15.200 --> 00:29:27.500
I have factors of 16, which are 1 and 16, 2 and 8, and 4 and 4; and I need factors of 16 that will add up to -8.
00:29:27.500 --> 00:29:42.500
And I can see that this one is correct, (x - 4) (x - 4), because that is going to give me a middle term of -8x.
00:29:42.500 --> 00:29:52.600
OK, this second set of factors is going to be x and x, and everything is positive.
00:29:52.600 --> 00:30:06.200
And factors of 4 that would add up to 4 would be 2 and 2: so, x² + 2x + 2x (that is going to give me 4x) + 4.
00:30:06.200 --> 00:30:14.200
Finally, I have a negative here; so I am going to do a plus here and a negative here.
00:30:14.200 --> 00:30:23.700
I want factors of 8 that add up to -2: well, 1 and 8 are too far apart, so I have 2 and 4.
00:30:23.700 --> 00:30:28.100
And I want it to be a -2, so I am going to make the 4 negative.
00:30:28.100 --> 00:30:32.600
This is going to give me a 2 here and a 4 here.
00:30:32.600 --> 00:30:43.800
OK, I could actually rewrite this, also, as (x - 4)²; and I am going to rewrite this as (x + 2)².
00:30:43.800 --> 00:30:49.400
So, when I look for my LCM, I am going to look for each factor that I have.
00:30:49.400 --> 00:30:57.900
And the first one is (x - 4), and the highest power I have it to is 2; I have an (x - 4) here, but this is really only to the first power.
00:30:57.900 --> 00:31:05.300
So, I am going to write this as (x - 4)².
00:31:05.300 --> 00:31:08.700
I took care of that factor that is right here, also.
00:31:08.700 --> 00:31:14.400
Now, I have another unique factor of (x + 2).
00:31:14.400 --> 00:31:19.400
The highest power in any one polynomial that I find it in is squared; I have an (x + 2) here,
00:31:19.400 --> 00:31:24.100
but it is to a lower power, so I am not going to worry about that; it is covered under this.
00:31:24.100 --> 00:31:31.300
The least common multiple of these three polynomials is (x - 4)² (x + 2)².
00:31:31.300 --> 00:31:37.100
Factor out each polynomial, and then take the product of their factors and use the power
00:31:37.100 --> 00:31:45.300
that is the highest power that any factor is present in, in one of the polynomials.
00:31:45.300 --> 00:31:50.600
Here is addition: adding rational expressions with different denominators.
00:31:50.600 --> 00:31:55.400
The first step is to get a common denominator.
00:31:55.400 --> 00:32:05.500
To achieve that, I am going to factor the denominators.
00:32:05.500 --> 00:32:09.800
There is a negative here, so this is plus; this is minus.
00:32:09.800 --> 00:32:22.800
I need factors of 6 that add up to -1; I am going to look at these two, and if I make the 3 negative, then I am going to get the correct middle term.
00:32:22.800 --> 00:32:26.600
So, I am going to put the 3 here and the 2 here.
00:32:26.600 --> 00:32:32.400
OK, so I factored out that first denominator; let's look at the second one.
00:32:32.400 --> 00:32:36.600
Leave the numerator alone for now, and concentrate on the denominator.
00:32:36.600 --> 00:32:45.400
I can see, pretty quickly, that I have a common factor of 4; so that will become x, and this will become minus 3.
00:32:45.400 --> 00:32:58.100
So, this is what I want to add; and I need to find the LCD, so I am going to look at all of these factors that I have in the denominator and find their product.
00:32:58.100 --> 00:33:04.400
I have (x + 2), and that only appears once, so I just leave it as the first power.
00:33:04.400 --> 00:33:14.300
I have (x - 3) here and here, and the highest number of times it appears is once and once; so it is (x - 3).
00:33:14.300 --> 00:33:26.700
Over here, I also have a 4; so the LCD...I am going to rewrite this with a 4 in front...is going to be 4(x + 2) (x - 3).
00:33:26.700 --> 00:33:34.600
Now, I need to rewrite these as equivalent fractions with this denominator.
00:33:34.600 --> 00:33:37.800
I look at the denominator, and I see what I am lacking.
00:33:37.800 --> 00:33:45.200
In this first denominator, I have (x + 2) (x - 3), but I am lacking a 4.
00:33:45.200 --> 00:33:56.000
So, I am going to multiply both the numerator and the denominator by 4 to form an equivalent fraction.
00:33:56.000 --> 00:34:08.500
This is going to give me 4(2x - 3)/(x + 2)(x - 3).
00:34:08.500 --> 00:34:21.400
Here, I am going to end up with...let's work with this one right down here...(3x² - 2x)/(4(x - 3)).
00:34:21.400 --> 00:34:25.000
What is lacking from the denominator? (x + 2)
00:34:25.000 --> 00:34:29.700
I have my 4; I have my (x - 3); I am lacking an (x + 2).
00:34:29.700 --> 00:34:37.900
So, I am going to multiply both the numerator and the denominator by that.
00:34:37.900 --> 00:34:53.400
This is going to give me (x + 2)(3x² - 2x), all over...oops, I need a 4 down there, as well;
00:34:53.400 --> 00:34:59.700
this 4 should be over there...times 4, times (x - 3)(x + 2).
00:34:59.700 --> 00:35:06.200
OK, I have a common denominator--it is a slightly different order that I wrote it in, but I still have a 4, an (x + 2), and an (x - 3).
00:35:06.200 --> 00:35:13.500
Now, I can add these: I am going to go ahead--I have my equivalent fractions, and I am going to add these.
00:35:13.500 --> 00:35:32.800
4(2x - 3) divided by 4(x + 2) (x - 3), plus (x + 2)(3x² - 2x) over this common denominator,
00:35:32.800 --> 00:35:38.900
4...I am going to write this in the same order as I wrote this one...(x + 2) (x - 3).
00:35:38.900 --> 00:35:59.900
Now, once you have a common denominator, all you need to do is add the numerators and put these over the common denominator.
00:35:59.900 --> 00:36:10.300
And the common denominator here is 4(x + 2) (x - 3).
00:36:10.300 --> 00:36:16.200
Here, this is all factored out; I need to add this and then see what I have an if there are common factors.
00:36:16.200 --> 00:36:19.200
I need to go ahead and multiply this all out.
00:36:19.200 --> 00:36:37.300
In the numerator, this is 4 times 2x (that is 8x) minus 12; here I have x times 3x², gives me 3x³.
00:36:37.300 --> 00:36:52.000
x times -2x gives me -2x²; I took care of this and this using the distributive property.
00:36:52.000 --> 00:37:00.500
2 times 3x² is 6x²; 2 times -2x is -4x.
00:37:00.500 --> 00:37:06.300
This is all, again, over that common denominator.
00:37:06.300 --> 00:37:11.400
Now, I can do a little more simplifying, because I can add like terms.
00:37:11.400 --> 00:37:17.200
My denominator is taken care of; let's look at this numerator.
00:37:17.200 --> 00:37:22.800
Starting with the largest power: I only have one x³ term, so that is 3x³.
00:37:22.800 --> 00:37:29.600
For x² terms, I have 6x² - 2x²; that is going to give me 4x².
00:37:29.600 --> 00:37:46.000
For x terms, I have...let's see...8x - 4x; that is going to give me 4x; for constants, I have -12.
00:37:46.000 --> 00:37:49.500
Now, what I am left with is this rational expression.
00:37:49.500 --> 00:37:56.000
And in order to simplify this, the way that you have to go about it is to use synthetic division.
00:37:56.000 --> 00:38:02.200
And there actually turn out to be no common factors, so I am going to leave it as it is.
00:38:02.200 --> 00:38:11.700
But you could check it by synthetic division; and the way to proceed would be to use synthetic division, and to divide this polynomial by (x + 2).
00:38:11.700 --> 00:38:19.000
And remember from the remainder theorem: if the remainder is 0, then (x + 2) is a common factor, and you would be able to cancel that out.
00:38:19.000 --> 00:38:26.300
I know 4 is not a common factor; I could also use synthetic division to determine that (x - 3) is not a factor of this.
00:38:26.300 --> 00:38:32.000
But if you did have common factors, then the final step would be to cancel those out.
00:38:32.000 --> 00:38:41.400
So, this was a pretty complicated problem; but proceeding as usual, factor the denominators, finding the least common denominator.
00:38:41.400 --> 00:38:57.500
Then convert to equivalent fractions by multiplying the numerator and denominator by whatever was lacking from the denominator to form the LCD.
00:38:57.500 --> 00:39:02.700
And this was 4 before, times this first rational expression.
00:39:02.700 --> 00:39:09.000
Down here, I had to multiply the second one times (x + 2), divided by (x + 2).
00:39:09.000 --> 00:39:18.200
I ended up with these two, with a common denominator; I rewrote them up here, and then I just added the numerators and did simplification.
00:39:18.200 --> 00:39:24.900
OK, in this example, we are going to be subtracting rational expressions with different denominators.
00:39:24.900 --> 00:39:29.100
The first thing to do is factor out the denominators to find the LCD.
00:39:29.100 --> 00:39:36.400
Factoring the first one: (x - 7) divided by...I am going to have an x here and an x here...
00:39:36.400 --> 00:39:41.000
Now, this is a negative sign; so I am going to have + and -.
00:39:41.000 --> 00:39:56.800
Factors of 12 are 1 and 12, 2 and 6, 3 and 4; factors of 12 that would add up to -4 would be these two, 2 - 6.
00:39:56.800 --> 00:40:03.800
So, I am going to put my 2 by the positive sign, and the 6 by the negative sign.
00:40:03.800 --> 00:40:08.800
And I am going to leave some space here for when I convert these to equivalent fractions.
00:40:08.800 --> 00:40:17.500
I am going to put my negative sign right here; and this is going to give me 2x + 3, and I am going to go ahead and factor this.
00:40:17.500 --> 00:40:22.400
This one is a little bit more complicated to factor, because the leading coefficient is not 1.
00:40:22.400 --> 00:40:26.200
So, I am going to get 2x here and an x here.
00:40:26.200 --> 00:40:31.400
Now, I have a negative sign here, but I have a positive sign in front of the constant.
00:40:31.400 --> 00:40:40.400
And that tells me that I am working with a negative and a negative, which will give me a positive here and a negative here.
00:40:40.400 --> 00:40:45.000
This is more complicated, because I have to take into account this 2.
00:40:45.000 --> 00:40:56.000
So, let's think about factors of 18, first of all, which are 1 and 18, 2 and 9, and 3 and 6.
00:40:56.000 --> 00:41:00.800
Now, when I am working with a leading coefficient other than 1, I like to start out with smaller numbers,
00:41:00.800 --> 00:41:05.800
because the one that is being multiplied with the 2 is going to become large.
00:41:05.800 --> 00:41:11.500
So, let's start out with 3 and 6 and look at different combinations.
00:41:11.500 --> 00:41:22.300
If I have 2x (let's put the 3 first) - 3, times (x - 6), that is going to give me 2x²;
00:41:22.300 --> 00:41:30.500
the outer terms give -12x; the inner terms give -3x; and the last terms give 18.
00:41:30.500 --> 00:41:39.000
Since -12x and -3x add up to -15x, this is the correct factorization.
00:41:39.000 --> 00:41:49.800
OK, now thinking about the LCD (least common denominator): I look at the factors I have, and I have an (x + 2).
00:41:49.800 --> 00:41:55.700
And it is only present once, so it is just a power of 1.
00:41:55.700 --> 00:42:04.800
I also have (x - 6), and I have one here, and it is only present once here; so again, I am just going to represent that once.
00:42:04.800 --> 00:42:10.500
And so, this one is taken care of; over here, I also have (2x - 3).
00:42:10.500 --> 00:42:16.500
Therefore, the LCD is going to be (x + 2) (x - 6) (2x - 3).
00:42:16.500 --> 00:42:19.600
Now, I need to convert these to equivalent fractions.
00:42:19.600 --> 00:42:27.400
To do that, I am going to multiply the numerator and the denominator by what is lacking from the denominator.
00:42:27.400 --> 00:42:39.600
Here, the factor that is missing is 2x - 3, so I need to multiply both the numerator and the denominator by that.
00:42:39.600 --> 00:42:50.400
Over here, I have the (2x - 3); I have (x - 6); but I need to multiply both the numerator and the denominator by (x + 2).
00:42:50.400 --> 00:42:54.100
Now, when I multiply these out, I am going to end up with a common denominator.
00:42:54.100 --> 00:43:15.200
I am going to take care of that multiplication: (x - 7) times (2x - 3), over (x + 2) times (x - 6) times (2x - 3);
00:43:15.200 --> 00:43:31.400
minus this entire second fraction, which is going to be (2x + 3) times (x + 2), divided by the common denominator.
00:43:31.400 --> 00:43:49.100
I am going to go ahead and write this in the same order as this one: (x + 2) first; (x - 6); and then (2x - 3).
00:43:49.100 --> 00:43:52.200
OK, I am just checking to make sure that I have everything accounted for.
00:43:52.200 --> 00:44:04.400
Now that I have a common denominator, I can subtract; so this is going to become (x - 7) times (2x - 3).
00:44:04.400 --> 00:44:26.600
And I need to be careful with the signs; it is going to be subtracting, so this whole thing is going to be the opposite signs...over the common denominator.
00:44:26.600 --> 00:44:33.200
OK, what is left is to simplify: this is already factored out, but I need to multiply this out,
00:44:33.200 --> 00:44:39.900
add together the like terms, and then factor and see if I can simplify.
00:44:39.900 --> 00:44:59.000
So, starting out right here, this is x times 2x; that gives me 2x²; x times -3...that is -3x;
00:44:59.000 --> 00:45:11.400
here I get -14x (that is -7 times 2x); and then -7 times -3 is 21.
00:45:11.400 --> 00:45:31.900
OK, minus what is in here: so, 2x times x is 2x²; 2x times 2 is 4x; now, the second term:
00:45:31.900 --> 00:45:49.200
3 times x is 3x; 3 times 2 is 6; all over the common denominator, (x + 2) (x - 6) (2x - 3).
00:45:49.200 --> 00:45:57.500
The next thing to do is take care of these signs: this equals 2x²...and do some simplifying.
00:45:57.500 --> 00:46:11.800
-3x and -14x is -17x, plus 21; here I am going to have a negative; that gives me -2x².
00:46:11.800 --> 00:46:20.400
Inside here, I have 4x and 3x, so that is 7x; but I need to take the negative of that--the opposite; this is actually -7x.
00:46:20.400 --> 00:46:30.500
For the constant I have 6, and the opposite of that is -6, all over the common denominator.
00:46:30.500 --> 00:46:33.500
Now, I have some like terms that can be combined.
00:46:33.500 --> 00:46:46.300
So, I am going to go down here; and this gives me 2x² - 2x²; these cancel, so the x² terms are gone.
00:46:46.300 --> 00:47:14.500
I have -17x and -7x combined, to give -24x; that leaves me with the constants: 21 - 6 is 15...over the common denominator.
00:47:14.500 --> 00:47:26.100
OK, looking at this, I can see that I don't have any common factors, so I can't simplify.
00:47:26.100 --> 00:47:36.000
You could pull a 3 from up here--you could factor out a 3; but that is not going to leave you with any common factors.
00:47:36.000 --> 00:47:42.800
There is no (x + 2), (x - 6), or (2x - 3) that is going to be left behind; so I can just leave this as it is.
00:47:42.800 --> 00:47:46.800
This was a pretty complex problem--pretty lengthy.
00:47:46.800 --> 00:47:49.300
Since it is subtraction, you have to be careful with the signs.
00:47:49.300 --> 00:47:57.900
Again, you are factoring the denominators of both, finding the LCD right here, then converting to equivalent fractions.
00:47:57.900 --> 00:48:06.000
This first fraction was lacking the (2x - 3) in the denominator; so I multiplied both by that.
00:48:06.000 --> 00:48:12.100
The second fraction needed an (x + 2) multiplied by both the numerator and the denominator.
00:48:12.100 --> 00:48:18.000
Once I did that, then it was a matter of subtracting, and I had the same denominator.
00:48:18.000 --> 00:48:26.400
I had to do some multiplying, combining, and simplifying to end up with the final answer that I have here.
00:48:26.400 --> 00:48:30.800
OK, here I have a complex fraction; and not only is it a complex fraction,
00:48:30.800 --> 00:48:37.600
but the rational expressions that we see are being added or subtracted in the numerator and the denominator.
00:48:37.600 --> 00:48:41.700
Again, the way to handle this is to handle the numerator and the denominator separately.
00:48:41.700 --> 00:48:53.700
So, first the numerator: I want to subtract this.
00:48:53.700 --> 00:49:03.000
The common denominator: well, my LCD is going to be x²y³.
00:49:03.000 --> 00:49:12.200
I look at what I have and what is lacking: well, what is lacking from this denominator, x², is a y³.
00:49:12.200 --> 00:49:23.000
So, I am going to multiply both the numerator and the denominator by y³, minus 1/y³.
00:49:23.000 --> 00:49:31.700
What is lacking here from the denominator is the x², so I am multiplying both the numerator and the denominator by x².
00:49:31.700 --> 00:49:48.400
This is going to give me y³ right here, over x²y³, minus x² over x²y³.
00:49:48.400 --> 00:49:53.700
Since I now have a common denominator, I can then subtract.
00:49:53.700 --> 00:49:59.900
So, it is y³ - x², over this common denominator.
00:49:59.900 --> 00:50:06.000
This is my numerator; so I am going to go over here and write this new numerator that I have.
00:50:06.000 --> 00:50:10.600
And this is much easier to work with, because now I just have a fraction.
00:50:10.600 --> 00:50:16.200
I have a rational expression; I don't have two rational expressions and subtraction.
00:50:16.200 --> 00:50:27.100
The denominator: that was the numerator--let's now work with the denominator.
00:50:27.100 --> 00:50:34.500
I am adding, and I am being asked to add (y/x³) + (x/y²).
00:50:34.500 --> 00:50:51.600
The LCD is x³y²; to convert this, I am going to have to multiply this fraction
00:50:51.600 --> 00:51:04.900
by y²/y², because that is what is lacking.
00:51:04.900 --> 00:51:12.000
And I am going to add that to x/y²; and what is lacking from this denominator is x³.
00:51:12.000 --> 00:51:16.900
I multiply this; this is x³ over x³.
00:51:16.900 --> 00:51:25.700
This gives me y times y² (is y³), over the common denominator, x³y²,
00:51:25.700 --> 00:51:42.300
plus x times x³ (is going to give me x⁴), over x³y².
00:51:42.300 --> 00:51:51.800
Since these now have a common denominator, I am going to add y³ + x⁴, over x³y².
00:51:51.800 --> 00:51:57.700
OK, I handled this as two different problems: a subtraction problem up here to get the numerator,
00:51:57.700 --> 00:52:03.000
and an addition problem adding rational expressions down here, to find the denominator,
00:52:03.000 --> 00:52:13.800
which is y³ + x⁴, over x³y².
00:52:13.800 --> 00:52:19.900
Once I am to this point, I just use my usual rules for dividing rational expressions.
00:52:19.900 --> 00:52:27.300
So remember: we are going to rewrite this as a division problem, because this fraction bar is just telling me to divide.
00:52:27.300 --> 00:52:38.600
This is going to give me...I am going to rewrite this down here as (y³ - x²), divided by x³y².
00:52:38.600 --> 00:52:51.500
This entire rational expression is being divided by this one: y³ + x⁴ divided by x³y².
00:52:51.500 --> 00:53:03.900
Dividing one rational expression by another is simply multiplying the first by the inverse of the second.
00:53:03.900 --> 00:53:14.500
So, I am going to rewrite this as x³y², divided by y³ + x⁴.
00:53:14.500 --> 00:53:18.100
Now, multiplication: the next step is always to simplify.
00:53:18.100 --> 00:53:28.600
So, let's look for common factors: I have x³ here and x² down here--get rid of the x²;
00:53:28.600 --> 00:53:32.700
this becomes x, because I took out that factor of x².
00:53:32.700 --> 00:53:41.700
I have a y² here and a y³ here; that cancels out, and this just becomes y, because I took out a y².
00:53:41.700 --> 00:53:44.400
Now, let's see what I have left and multiply that.
00:53:44.400 --> 00:53:59.000
I have an x, times this whole thing, which is y³ - x², divided by...
00:53:59.000 --> 00:54:05.700
I have a y left here, and I have y³ + x⁴.
00:54:05.700 --> 00:54:11.900
And now, I have simplified it as far as I can.
00:54:11.900 --> 00:54:16.800
And this took many steps; you have to be careful and make sure that you keep track of everything.
00:54:16.800 --> 00:54:24.200
But start out by simplifying the numerator, by subtracting to get this numerator.
00:54:24.200 --> 00:54:31.900
Simplify the denominator by adding to get this for the denominator.
00:54:31.900 --> 00:54:42.700
Then, treat this as a regular complex fraction, where we are going to take this numerator and divide by the denominator.
00:54:42.700 --> 00:54:49.000
And we handle that by taking the first rational expression and multiplying by the reciprocal of the second.
00:54:49.000 --> 00:54:54.900
I found common factors; I canceled those out; and this is what I ended up with.
00:54:54.900 --> 00:54:57.800
And it cannot be simplified any more.
00:54:57.800 --> 00:55:02.900
That concludes this lesson on adding and subtracting rational expressions.
00:55:02.900 --> 00:55:04.000
Thanks for visiting Educator.com!