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# Least common multiple of polynomials

## Video transcript

so they're asking us to find the least common multiple of these two different polynomials so the first one is 3z to the third minus 6t squared minus 9 Z and the second one is 7 Z to the 4th plus 21 is Z to the third plus 14 Z squared now if you're saying well what is the lead you're familiar with the 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 4 and 6 you literally could look at all the multiples and see which ones are which one is least so you go 4 8 12 16 so on and so forth you could do the same thing for 6 you go 6 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 is to actually factor these numbers out we could view for you could view 4 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 4 & 6 is going to be or it's going to be equal to what's going to have to have the factors of both of them so it's going to have to have two fours two times two 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 be in order to be divisible by 6 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 of them we have to have at least two twos and we have to have at least one 3 because the two twos take care of this one two 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 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 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 to the third minus 6z squared minus 9 z let's see immediately let's see all of these terms are divisible by 3 Z so let's factor out a 3 z so it's 3 Z times Z squared if you factor out a 3 Z out of that see it's going to be minus 2z if you factor out a 3 Z out of that and then minus 3 and notice if you were to distribute this 3z 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 you multiply them we get negative 3 and if we add them we get negative 2 and one's going to be positive one's going to be negative since their product is negative so let's see it sounds like negative 3 and positive 1 so we could rewrite this as 3 Z times Z plus 1 times Z minus 3 I think I've factored this first this first polynomial is about as much as I can 1 times negative 3 is negative 3 1 Z minus 3 Z is negative 2 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 divisible by 7 Z squared so I could write this as 7z squared times Z squared when you factor out a 7z squared here you're just left with Z squared and then plus 21 divided by 7 is 3 Z to the third divided by Z squared is Z and then plus 14 divided by 7 is 2z squared divided by z squared is 1 so it's just going to be a 2 there and so this is going to be the same thing as 7z squared and this can be factored into see 2 times 1 is 2 2 plus 1 is 3 so Z plus 1 times Z plus 2 now let's think about the least common mo 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 two as simple 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 3z it's gotta contain and let me let me expand it out a little bit it's got to continue to 3 it's got to contain a Z it's got to contain a Z plus 1 z plus 1 it's I'm going to try the little dot there since I'm it's got to contain a Z plus 1 it's got to contain a Z minus 3 let's see it's got to contain a 7 we do not have a 7 here yet so we have to include 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 squared well we only have a Z right now so let's start in another Z so I could throw in another I could write set I could put a Z at front or I could just make this a squared it still contains that Z but now we contain another Z or multiplying another Z to have Z squared see we already have a Z plus 1 in here we need a Z plus 2 as well Z plus 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 21 Z squared times Z plus 1 times Z minus 3 times Z plus 2 si si si 2 and I write 6 Z plus 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 ok the least common multiple has to contain each of has to has to be a superset has to contain all of these but we don't want to contain you know I could multiply this x times a hundred 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 so I just wanted a common multiple I can multiply that times 100 1,200 would also be a multiple of four and six but it would be the least common multiple so we don't want to we don't want to do that hopefully found that interesting