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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 a polynomial with a
negative first term?
Well, the first step
in this process is to
determine what the negative GCF
is, and we'll walk through that
process in a moment, the second
step is to express each term as
a product with the negative GCF.
Thirdly we'll factor out the
negative GCF using what we call
reverse distribution and finally
we should always check our
factoring process with
multiplication.
so let's take a look
at an example.
If you met with in this
particular polynomial
the first term is negative
and so we'd like to
factor out that negative. So
that the first term will be
positive.  I've gone
ahead and set
each of my terms up separately
to determine the prime
factorization for each of those
terms so if we take -18X^4Y^3
that can also be
written as a product -1 times
well 18 is 9 times 2 is prime
however 9 is not so we'll
call that 3 times 3.
And we have four X's being
multiplied and three Y's
Moving to 6X^2 Y^2,
that can be written as a
product of 2 times 3 we have two
X's being multiplied in this
one and two Y's.
And finally -15X^3Y can be
written as -1 times 3 times 5
for -15. 3X is being
multiplied in and one Y.
Annd so once again
we'll determine
and identify all of the
factors that they have in
common so it looks like they
each have a factor of 3 they
each have two X's.  Our first
X and our second factor of X.
And finally they each have
one factor of Y at least so
our regular GCF could be written
as 3 times X times X times Y.
Which is 3X^2Y however we like
to take the
negative GCF which is just the
opposite so we have negative
3X^2Y being our negative GCF.
The next step is to express
each term is a product of that
negative GCF and the other
factors that are left over so
-18X^4 is the same thing as
-3X^2Y times.   Well, we want
to see that first-term negative
so we need to multiply that by a
positive 6X^2Y^2 to give us the
term -18X^4Y^3.
Moving along to the next one
we want to make a product
of negative 3X^2Y.
However 6X^2Y^2
 is a positive terms so
we need to multiply with a
another negative.  And that
would be negative times -2Y
to complete that term.   And
finally -15 X^3Y could
be written as  -3X^2Y  times
positive 5X so
we've expressed each of these
terms is a product of the
negative GCF and the factors
that are left over. Our next
step is to factor out the
negative GCF through
reverse distribution.
So, I bring the negative factor
out front and after dividing
that negative term, sorry,
negative monomials out of each
term were left over with 6X^2Y^2
a negative 2Y
land a positive 5X so
 -3X^2Y times 6X^2Y^2 -2Y+5X
is the factored form
of the polynomial
-18X^4Y^3+6X^2Y^2-15X^3Y.
We should always check
our work just to make
sure that we're safe
with what we did.
So -3X^2Y times
6X^2Y^2-2Y+5X.
 So we come back to good old
distribution here and we end up
getting -3Xtimes 6 which is -18X
X^2 times X^2 is X^4
Y times Y^2 is Y^3.
Moving to the next term
-3 times  -2 positive 6.   X^2
has no other X
 to multiply with in that term.
Y times Y is Y^2.
Moving to the last one
-3 times 5 is -15.
X^2 times X is X^3.
And we just have one Y
multiplying in to that final
term there so we can safely say
now that 
-3X^2Y times 6X^2Y^2-2Y+5X
is the factored form because
our polynomials match here.
If you have any other
questions regarding how to
factor out the negative
GCF from a polynomial
please contact your
Highlands instructor.
Thank you for watching.
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