Monomial Greatest Common Factor

u12_l1_t1_we1 Monomial Greatest Common Factor

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.