 This video will talk about adding and subtracting rational expressions. This is done by just, just like adding and subtracting fractions, it's just a little more complicated because you're going to have some variables involved. But when you, let's think about this problem. How would we add this thing? If you remember, we need a lowest common denominator, and that's the smallest number or quantity that goes into both numbers or quantities if we're talking about, um, rationals. And so we look at these numbers of fifteen and nine, and they have forty-five in common. So the least common denominator is going to be forty-five. And then remember that you just multiply to make both of those denominators forty-five. So to make this one forty-five, I would have to multiply by three. Three times fifteen would be forty-five, but it has to be three over three, a factor of one, so I don't change the value of my fraction. And this one over here, we would multiply by five because five times nine is forty-five, and it would actually be five over five. So then when we multiply, we have six on the top of the first fraction and forty-five on the bottom. And then one times five, again we're going to multiply here, and multiply down here, we're going to get five over forty-five. And then we just have to know how many forty-fifths do we have? We have six plus five more, which would be eleven forty-fifths. Now let's think about just least common denominators in practice. So we have six and we have four. One way to do this is to just start listing out all the multiples of six. So six and then twelve and then eighteen. And then we can list multiples of four. That would be four and eight and twelve, and now I have found the first number they have in common, so that would be the smallest thing they have in common. So the least common denominator here must be twelve. Now when we look here, we have to consider the number and the letters. So two and three would be two, four, six, and six, I know is common with three. So I'm going to realize that that would be my common factor here. You need two. It would be three, six. Oh, there it is. And then with the variable, we have to realize that we have to build up the denominators. So that makes it have to be the largest exponent. So we have a six for the number and we have x to the fourth, giving us six x to the fourth for the least common denominator. What do we do when we've got expressions? Well, we just factor. So we have x plus two and then over here, we would factor this one and it would be x plus two and x minus two. And I just need to know all the different factors so then I can make my denominators look the same. So if I look at this, I have x plus two and when I go to the second fraction, I've already got x plus two and if I compare their exponents so I could take the largest one, but they're the same, I don't need a second one and then I need the x minus two because that's a different factor that I haven't listed yet. So that x plus two, x minus two would be my least common denominator. Okay, one more. So we have to factor factors of negative 12 that allow it to negative one. That would be three and four with opposite signs. So we want a negative answer so it will be negative four and positive three. And then over here, we just have x plus three. That is the denominator there. So the least common denominator, I've got x minus four, I've got x plus three and when I go to my second fraction, I realize I've already listed that factor and they have the same exponent so I have x minus four and that is a three, x plus three. When we add or subtract rational functions, the first thing we want to do is factor those numerators and denominators separately. That way we can find the least common denominator. Then we'll multiply each fraction by that factor of one. Remember at the very beginning we had a five over five and three over three or whatever it was. So we need this factor that looks like a fraction but the same thing in both parts of it and that's made up of the missing parts of that fraction from the least common denominator. Then we're going to add or subtract the numerators only by combining like terms and when you subtract don't forget to distribute the negative and then reduce it if we have to. So when we did one earlier, we had x over six and x over four but we weren't adding, but we found out that the least common denominator was twelve. Just like we did before and we had numerical fractions, we need to multiply this one by two over two and we need to multiply the second fraction both times three over three. When we combine here and multiply we're going to get two times x on the top and two times six on the bottom will be twelve plus and combine three by multiplying three times x is three x and on the bottom four times three again will be twelve. My denominators are the same so I get to add. So I have two x plus three more x's so I have five x on the bottom and remember it said numerators only. So I carry along the twelve. I want to know how many twelfths I have. I have five x twelfths. We factor. This denominator has a common factor of x leaving me with x plus four and that denominator is already factored. So the least common denominator is going to be equal to this x and x plus four and I've already listed that x factor so I don't need to list it again. So my least common denominator is x times x plus four. So when I look at this fraction it's four over x times x plus four. It already has my least common denominator. So I just have to rewrite it in factored form and then minus and I have the x but I don't have the x plus four so I'm going to have to multiply the x plus four over x plus four. You can see there's my denominator, least common denominator of x and x plus four but on the top I have to distribute my nine so nine times x would be nine x and nine times four would be plus thirty six and then here's the tricky part. We are just subtracting and we are subtracting the whole numerator here. So if I were to write it out as one long fraction it would be four and then distribute it here it'd be minus nine x and distribute it here it'd be minus thirty six and this is the one that everybody misses over my x times x plus four and then I just have to simplify. I have two constants so four minus thirty six will give me a negative thirty two and then minus my nine x and I would need to look and see if I could factor that to possibly reduce this fraction but negative thirty two and negative nine x the only thing they have in common is negative one which won't reduce with the bottom so this is good alright here we go I think we did this one earlier too so this one was x minus four and x plus three and that makes the least common denominator x minus four first factor that I see and x plus three the next factor that I see and then the second fraction I've already listed that particular factor and it has the same exponent on it so we just have x minus four x plus three so once again my first fraction actually has the right denominator in it so I don't have to do anything to that fraction that rational expression gets to stay minus now I have the x plus three but I'm missing the x minus four so we have to multiply by the missing factor of the least common denominator and on the bottom then you can see that there's my x plus three and x minus four and it doesn't matter how you write those factors because we can multiply in any order so you look at those two denominators and I put those factors in the different order but they're the same thing so now I just have to distribute my two x to the x gives me two x squared and to the negative four gives me minus eight x and then I remember about this negative has to go into that whole numerator so I have five minus two x squared and then negative times a negative remember is I'm even going to put it in a different color plus eight x and then that's all over my denominator of x plus three x minus four and the only thing I have left to do then is combine my like terms if there are any and there aren't so I have negative two x squared and then plus eight x plus five I'm just going to write it in standard quadratic form so that I can possibly factor if I have to and I want to see if it's factorable so negative two times five would be negative ten and the middle is eight and I'm thinking of factors of negative ten that would add up to eight I only have one and ten and they're opposite signs so that won't be eight or two and five being opposite signs would give me three so this one cannot be reduced any further and there's our answer okay our final one here then x plus two is this denominator and over here I have x plus two and x minus two so the least common denominator is you guessed it x plus two I've already listed x plus two so I don't have to list it again and then x minus two the first fraction it has the x plus two but it doesn't have the x minus two so I have to multiply by x minus two over x minus two so distribute the six everywhere inside and I get in fact I'm going to read it over here x minus two over x minus two is what I have to multiply by and my fraction was six over x plus two and when I fix just this fraction I get six x minus twelve over that least common denominator and then I'm going to add and when I look at this fraction I don't have the least common denominator is my denominator so I don't have to do anything to it I just have x over x plus two x minus two and notice once again I wrote it in different order but it doesn't matter we just have to have them all listed down there and combining like terms I'm going to have to add those two x's so six x plus one more x will be seven x minus twelve on the top over x plus two x minus two and seven x minus twelve I'm looking to see if I can factor that at all but I can't they have nothing in common so there is my final answer