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Greatest common factor of monomials

Video transcript

find the greatest common factor of these monomials now the greatest common factor of anything is the largest factor that's divisible into both if we're talking about just pure numbers into both numbers or in this case into both monomials now we have to be a little bit careful when we talk about greatest in the context of algebraic expressions like this because its greatest from the point of view that it includes the most factors of each of these each of these monomials it's not necessarily the greatest possible number because maybe some of these variables could take on negative values maybe they're taking on values less than one so if you square it is actually going to become a smaller number but I think without getting too much into the weeds there I think if we just kind of run through the process of it you'll understand it a little bit better so to find the greatest common factor let's just essentially break down each of these numbers into what we could call their prime factorization but it's kind of a combination of the prime factorization of the numeric parts of the number plus essentially the factorization of the variable parts so if we wanted to write ten we could say or we if we were to write ten CD squared we can rewrite that as the product of the prime factors of ten the prime factorization of ten is just two times five those are both prime numbers so ten can be broken down as two times five see can only be broken down by C we don't know anything else that C can be broken into so C so two times five times C but then the d squared can be rewritten as D times D D times D this is what I mean by writing this monomial essentially as the product of its constituents for the numeric part of it it's the constituents are the prime factors and for the rest of it is we're just kind of expanding out the exponents now let's do that for 25 C to the third d squared so 25 right here that's 5 times 5 so this is equal to 5 times 5 and then C to the third that's times C times C times C and then d squared times D squared d squared is times D is d so what's their greatest common factor in this context well they both have at least one v they both have at least one v then they both have at least one C over here so let's just take up one of the C's right over there and then they both have two DS they both have two DS so the greatest common factor in this context the greatest common factor of these two monomials is going to be the factors that they have in common so it's going to be equal to this five times we only have one C in common times and we have two DS in comments times D times D so this is equal to five c d squared and so five d squared we can kind of view it as the greatest but I'll put that in quotes where you know depending on whether C is negative or positive and D is a greater than or less than zero but this is the greatest common factor of these two monomials it's divisible into both of them and it uses the most factors possible