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Welcome to the Georgia Highlands
College Math 97 and Math 99
tutorial videos.   In
this video segment we'll be
answering the question how do
you factor polynomial using GCF?
Well, the first step in this
process is to determine what the
GCF is between all of the terms,
the second step in this process
is to express each term as a
product with the GCF the third
step is to actually factor out
the GCF which you might want to
refer to as reverse distribution
and finally we can check our
work by multiplying. Let's
take a look at an example.
Let's take a look at
the example problem
27X^2Y^3-9XY^2+81XY.
So we'd like to factor this
trinomial using the GCF method.
The first step in the GCF
method is to determine the GCF
for all the terms.  Now you may
be able to look at this and take
out all of the factors that they
have in common but if not you
can always use the method for
finding the greatest common
factor. So off to the side
here we'll look and find what
that greatest common factor is.
So we first have the term
27X^2Y^3.
Which can also be written as
3 times 3 times 3 which is 27.
X times X, X^2; times
Y times Y times Y.
Which gives us our Y^3.
Our next term is negative
9XY^2.  Weill that could also
be written as -1 times 3 times
3 times X times Y times Y.
And our final term 81XY
can be written in its
prime factors form as
3 times 3 times 3 times 3 times
X times Y and once again
we'd like to go back and
identify the factors
that these have in
common.   So, they all
have a factor of two 3's they
each have one X.and they each
have a factor of Y.
So we have 3 times 3 times X
times Y which gives us
a GCF of 9XY.   Meaning that
they each have a factor of 9XY
within that term.   Our next
step in the process is to
express each of these terms as a
product of the GCF and whatever
other factors are within the
term.   So if we  look at the
first-term 27X^2Y^3 we
can also call that 9XY times
well 9 times 3 is 27.  X times
X is X^2 and Y times Y^2
is Y^3.  So, our first term is
completely written as a product
of the GCF and then the other
factors within that term our
next term negative 9XY^2
can be written as positive 9XY
there's our GCF and then we have
to figure out what we had to
multiply to make that term.  So
that would be multiplied with a
-1 we already have an
X there,  so we don't need to
multiply with another X but we
do need to multiply with another
Y to make Y^2.   And
finally our last term 81XY
can be written as 9XY times
9 to make 81XY.   So we've
completed the second step of
expressing each term as a
product of the GCF and their
other factors. Our third step
is to do reverse distribution.
So instead of multiplying
into a set of
parentheses we're going to
divide that GCF off of each
term and pull it out front.
So, if I bring my GCF out
front, I'm eft over with
the trinomial 3XY^2-1Y,
but we don't write the
1,  it's understood to be there.
We simply write Y and then +9.
This is the completely factored
form of the polynomial 
27XY^2Y^3-9XY^2+81XY.
And the reason that
this is considered factored is
because it answers that question
what did I have to multiply
together to get that
expression. We can check this,
which is our final step
of the process, by simply
multiplying it back out and
seeing if we end up what we
began with.   So, if I multiply
9XY by 3XY^2-Y+9.  I will
distribute that 9XY to all three
of the terms within within my
parentheses 9 times 3 is
27; X times X is X^2.
Y times Y^2 is
Y^3. so 27X^2Y^3.
Moving to the next term
9XY times negative Y
well a positive times
the negative is a negative.
9 has no other number
to multiply with except
understood 1 so it remains 9.
X has no other X's to multiply
with.   Y times Y is Y^2.  
Moving to the last term,
we have 9XY times 9 which
gives us positive 81XY which
is exactly the polynomial that
we began with. Therefore,  we
can now be confident in saying
that 9XY times 3XY^2-Y+9
is the factored form
of the polynomial
27X^2Y^3-9XY^2+81XY.
Thank you for watching
this tutorial video.
if you have any other
questions about how to factor
using the GCF method,
please contact your
Highlands instructor.
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