Least common multiple of polynomials | Mathematics III | High School Math | Khan Academy

So they're asking us to find the least common multiple of these two different polynomials. The first one is (3z^3r - 6z^2 - 9Z) and the second one is (7Z^4 + 21Z^3r + 14z^2).

Now, if you're saying, well, what is the LCM? 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 4, 8, 12, 16, so on and so forth. You could do the same thing for six; you could go six, 12, 18, 24, so on and so forth. And you immediately see that they 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 as being (2 \times 2) if you look at its prime factorization, and 6 is (2 \times 3). So you could say that the least common multiple, the LCM, of four and six is going to have to have the factors of both of them.

So it's going to have to have two (2)s, (2 \times 2). It's going to have to have a (2) and a (3). Well, we already have a (2); in fact, we have two of them. So in order to be divisible by (3), we have to have (3) as one of the factors.

And so, when you look at it that way, you say, hey, we have to contain all of the factors of each of them. We have to have at least two (2)s and we have to have at least one (3), because the two (2)s take care of this one (2) 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 going to think about it a little bit more. It's essentially the same idea, but we're going to think about it a little bit more with the second lens where we're going to 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 want to find just any multiple; we want to find the least common multiple. So let's factor each of them.

So this first one (3Z^3r - 6z^2 - 9Z), let's see immediately, all of these terms are divisible by (3Z). So let's factor out a (3Z). So it's (3Z \times (z^2ar - 2Z - 3)). If you factor out a (3Z) out of that, you see it's going to be (z^2ar - 2) if you factor out a (3Z) out of that, and then minus (3).

Notice, if you were to distribute this (3Z) back, you would get exactly what we have up here. Can we factor this further? Can we think of two numbers that, if you multiply them, we get (-3) and if we add them, we get (-2)?

One's going to be positive, one's going to be negative, since their product is negative. So, it sounds like (-3) and (+1), so we could rewrite this as (3Z \times (Z + 1)(Z - 3)). I think I've factored this first polynomial about as much as I can.

Now, let's factor this other polynomial over here, this fourth-degree polynomial. Every one of those terms looks like they're divisible by (7z^2). So I could write this as (7z^2 \times (z^2 + 3Z + 2)).

When you factor out a (7z^2) here, you're just left with (z^2) plus (21) divided by (7) is (3Z); the (3) at (z^2) is (z) and then plus (14) divided by (7) is (2).

So this is going to be the same thing as (7z^2), which can be factored into ( (Z + 1)(Z + 2)). 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 as simple expressions as we will find useful.

The least common multiple of these two things has to contain each of these factors. So the least common multiple has to contain a (3Z); it's got to contain, and let me expand it out a little bit: it's got to contain a (3), it's got to contain a (Z), it's got to contain a (Z + 1).

I don't have to write a little dot there since it's got to contain a (Z + 1); it's got to contain a (Z - 3). Let's see, it's got to contain a (7) — we do not have a (7) here yet, so we have to include a (7).

So I'll put the 7 out front with the numbers; it's got to include a (7), it's got to include a (Z^2). Well, we only have a (Z) right now, so let's throw in another (Z).

So I could throw in another; 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 multiply by another (Z) to have (Z^2).

See, we already have a (Z + 1) in here; we need a (Z + 2) as well. (Z + 2) 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 going to be (21z^2 \times (Z + 1)(Z - 3)(Z + 2)). I say two and I write (6z + 2), 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 or a very similar thing than what we're doing when we're finding least common multiples of regular numbers. We're looking at their factors and, in the case of numbers, prime factors.

And then we say, okay, the least common multiple has to contain, it has to be a superset, has to contain all of these, but we don't want to contain, you know, I could multiply this times (100); it's still going to be a common multiple of these two, but it's no longer the least common multiple.

Likewise, (12) is the least common multiple of (4) and (6). If I just wanted a common multiple, I could multiply that times (100); (1200) would also be a multiple of (4) and (6), but it wouldn't be the least common multiple.

So we don't want to do that. Hopefully, you found that interesting.