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 just talking about 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
for each of these monomials,
it's not necessarily the greatest possible number
because maybe some of these variables
can take on negative values;
maybe they are taking on values less than one
so if squared they actually 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 part.
So if we wanted to write 10,
or if we wanted to write 10cd^2
we can rewrite that
as the product of the prime factors of 10
- which is just 2 * 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^2 can be rewritten
as d times d.
This is what I mean
by writing this monomial
as the product of its constituants.
For the numeric part of it,
it's the constituants of the prime factors
and for the rest of it
we are just expanding out the exponents.
Now, let's do this for
25 c to the third, d squared.
So 25, that is 5 * 5.
So this is equal to 5 * 5.
And then c^3, that is
c times c times c.
And then d-squared,
that is d times d.
So what is their greatest common factor
in this context?
Well, they both have at least one 5,
and they both have at least one c
and then they both have two d's.
So the greatest common factor in this context
the greatest common factor
of these two monomials,
will be the factors that they have in common.
It will be equal to
this five, times
we only have one c in common,
times - we have two d's in common.
So this is equal to
5 times c times d-squared.
So 5 c d-squared
we can view as the greatest --
I'll put that in quotes,
you know, depending on whether c
is negative or positive -
and d is greater than
or less than zero.
But this is the "greatest" common factor
of these two monomials.
It's devisable into both of them
and it uses
the most factors possible.