Subtracting rational expressions: factored denominators

- [Voiceover] Pause this video and see if you can subtract this magenta rational expression from this yellow one. Alright, now let's do this together. And the first thing that jumps out at you is that you realize these don't have the same denominator and you would like them to have the same denominator. And so you might say, well, let me rewrite them so that they have a common denominator. And a common denominator that will work will be one that is divisible by each of these denominators. So it has all the factors of each of these denominators and lucky for us, each of these denominators are already factored. So let me just write the common denominator, I'll start rewriting the yellow expression. So, you have the yellow expression, actually, let me just make it clear, I'm going to write both, the yellow one, and then you're going to subtract the magenta one. Whoops. I'm saying yellow but drawing in magenta. So you have the yellow expression which I'm about to rewrite, actually, I'm going to make a longer line, so the yellow expression minus the magenta one, minus the magenta one, right over there. Now, as I mentioned, we want to have a denominator that has all, the common denominator has to have, be divisible by both, this yellow denominator and this magenta one. So it's got to have the Z plus eight in it. It's got to have the 9z minus five in it. And it's also got to have both of these. Well, I already, we already accounted for the 9z minus five. So it has to have, be divisible by Z plus six. Z plus six. Notice just by multiplying the denominator by Z plus six, we're not divisible by both of these factors AND both of these factors because 9z minus five was the factor common to both of them. And if you were just dealing with numbers when you were just adding or subtracting fractions, it works the exact same way. Alright, so what will the numerator become? Well, we multiply the denominator times Z plus six, so we have to do the same thing to the numerator. It's going to be negative Z to the third times Z plus six. Now let's focus over here. We had, well, we want the same denominator, so we can write this as Z plus eight Z plus eight times Z plus six, times Z plus six times 9z minus five. And these are equivalent. I've just changed the order that we multiply in it, that doesn't change their value. And if we multiplied the, so we had a three on top before and if we multiply the denominator times Z plus eight, we also have to multiply the numerator times Z plus eight. So there you go. And so, this is going to be equal to, this is going to be equal to, actually, I'll just make a big line right over here. This is all going to be equal to. We have our, probably don't need that much space, let me see, maybe that, maybe about that much. So I'm going to have the same denominator and I'll just write it in a neutral color now. Z plus eight times 9z minus five times Z plus six. So over here, just in this blue color, we want to distribute this negative Z to the third. Negative Z to the third times Z is negative Z to the fourth. Negative Z to the third times six is minus 6z to the third. And now this negative sign, right over here, actually, instead of saying negative Z, negative of this entire thing, we could just say plus the negative of this. Or another of thinking about it, you could view this as negative three times Z plus eight. So we could just distribute that. So let's do that. So negative three times Z is negative 3z and negative three times eight is negative 24. And there you go. We are, we are done. We found a common denominator. And once you have a common denominator, you could just subtract or add the numerators, and instead of doing this as minus this entire thing, I viewed it as adding and then having a negative three in the numerator, distributing that and then these, I can't simplify it any further. Sometimes you'll do one of these types of exercises and you might have two second-degree terms or two first-degree terms or two constants or something like that and then you might want to add or subtract them to simplify it but here, these all have different degrees so I can't simplify it any further and so we are all done.