Adding & subtracting rational expressions
Least common multiple of polynomials
- [Voiceover] So they're asking us to find the least common multiple of these two different polynomials. So the first one's three z to the third minus six z squared minus nine z and the second is seven z to the fourth plus 21 z to the third plus 14 z squared. Now, if you're saying, "Well, what is a," you're familiar with least common multiple of two numbers. One way to think about them is if I were to find, say, the least common multiple between, I don't know, four and six, you literally could look at all the multiples and see which one is least. So you go four, eight, 12, 16, so on and forth. You're gonna do the same thing for six. You'd go six, 12, 18, 24, so on and so forth. And you immediately see that they do have, they'll actually have multiple common multiples, but the least of the common multiples you immediately see is going to be 12. Now another way to think about it is to actually factor these numbers out. We could view four, you could view four as being two times two if we look at its prime factorization. And six is two times three. So you could say that the least common multiple, the LCM, of four and six is going to be, well, it's gonna be equal to, we're gonna have to have the factors of both of them. So it's gonna have to have two fours, two times two. It's gonna have to have a two and a three. Well we already have a two. In fact, we have to them. So in order to be divisible by three, we have to be, in order to be divisible by six, we have to have three as one of the factors. And so when you look at it that way, you say, "Hey look, we have to contain "all of the factors of each," we have to have at least two twos and we have to have at least one three because the two twos take care of this one too right over there. And you see that this is also going to be equal to 12. Now when we think about it for polynomials, we're gonna think about it a little bit more. It's essentially the same idea, but we're gonna think about it a little bit more with the second lens. We're gonna think about the factors and say, well the least common multiple needs to contain the factors of both but it shouldn't contain more than that. You can always find a multiple of two polynomials by just multiplying them, but we don't wanna find just any multiple. We wanna find the least common multiple. So let's factor each of them. So this first one, three z to the third minus six z squared minus nine z. Let's see. Immediately, let's see, all of these terms are divisible by three z, so let's factor out of three z. So it's three z times z squared. We factor out of three z out of that. See, it's gonna be minus two z if we factor out of three z out of that. And then minus three. And notice, if you were to distribute this three z back, you would get exactly what we have up here. And so let's see, can we factor this further? Can we think of two numbers that if we multiply, we get negative three and if we add them, we get negative two? And one's gonna be positive, one's gonna be negative since the product is negative. So let's see. It sounds like negative three and positive one. So we can rewrite this as three z times z plus one times z minus three. I think I have factored this first polynomial about as much as I can. One times negative three is negative three. One z minus three z is negative two z, so that looks good. So now let's factor, now let's factor this other character over here, this fourth degree polynomial. So, every one of those terms look like they're visible by seven z squared, so I could write this as seven z squared times z squared. If we factor out of seven z squared here, you're just left with z squared. And then plus, 21 divided by seven is three, z to the third divided by z squared is z. And then plus, 14 divided by seven is two. z squared divided by z squared is one. So it's just gonna be a two there. And so this is gonna be the same thing as seven z squared. And this can be factored into, let's see, two times one is two, two plus one is three. So, z plus one times z plus two. Now let's think about the least common multiple. We've factored each of these just the way that we, when we did the prime factorization for regular numbers. Now we have factored this down to simpler expressions as we will find useful. And so, the least common multiple of these two things has to contain each of these factors. So the least common multiples got to contain a three z. It's got to contain, and let me expand it out a little bit. It's got to contain a three. It's got to contain a z. It's gotta contain a z plus one, z plus one. I don't have to write the little dot there. It's got to contain a z plus one. It's gotta contain a z minus three. Let's see. It's gotta contain a seven. We do not have a seven here yet, so we have to include a seven, so I'll put the seven out front with the numbers. It's gotta include a seven. It's gotta include a z squared. Well we only have a z right now, so let's throw in another z. So I could throw in another, I could write, I could put a z out front or I could just make this a squared. It still contains that z, but now we contain another z, or multiplying by another z to have z squared. See, we already have a z plus one in here. We need a z plus two as well, z plus two as well. And there you have it. This is the least common multiple. If I were to write it all out in a neutral color, it's gonna be 21 z squared times z plus one times z minus three times z plus two (chuckle) I say two and I write six. z plus two. And we are all done. And I really want you to appreciate. This is the exact same thing we're doing when we're doing, a very similar thing than what we're doing when we're finding these common multiples of regular numbers, looking at their factors. In the case of numbers, prime factors. And then we say, "Okay. The least common multiple "has to be a superset, has to contain all of these, "but we don't wanna contain," you know, I could multiply this times a hundred, it's still gonna be a common multiple of these two, but it's no longer the least common multiple. Likewise, 12 is the least common multiple of four and six. If I just wanted a common multiple, I can multiply that times a hundred, 1200 would also be a multiple of four and six, but it wouldn't be the least common multiple, so we don't wanna do that. Hopefully you found that interesting.