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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 it's greatest
from the point of view that it includes the most
factors of 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 1. So if you square
it, it's 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. If we were to
write 10cd squared, we can rewrite
that as the product of the prime factors of 10. The prime factorization
of 10 is just 2 times 5. Those are both prime numbers. So 10 can be broken
down as 2 times 5. c can only be broken down by c. We don't know anything else
that c can be broken into. So 2 times 5 times c. But then the d squared can
be rewritten as 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 of the prime factors. And for the rest
of it, we're just kind of expanding
out the exponents. Now, let's do that for 25c
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 times d. So what's their greatest
common factor in this context? Well, they both
have at least one 5. 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 d's. 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 5 times-- we only have one c in common, times--
and we have two d's in common, times d times d. So this is equal to 5cd squared. And so 5d squared, we can kind
of view it as the greatest. But I'll put that
in quotes depending on whether c is
negative or positive and d is greater
than or less than 0. 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.