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Welcome to the Georgia Highlands
College Math97 and Math99
tutorial videos. In this video
segment we'll be answering
the question: how do you
factor a polynomial using
grouping and rearranging terms?
Well, the first step in this
process is to actually
rearrange the terms so that
you're matching up terms
with common factors in them.
Secondly, we'll group those
terms with common factors.  
And then we'll factor the GCF
out of each of those groups.
Then we'll finish off by
factoring out the greatest
common binomial factor and
checking with multiplication.
Let's take a look at an example.
We have the example here:
XY-7-X+7Y.
And if you notice the first
two terms in this polynomial
don't seem to have any
factors in common and
neither do the last two
terms in the polynomial.
So we need to do some
rearranging before we
can begin grouping
these terms together.
And I notice here that I
have a factor of X in the
third term as well as in the
first and there is a factor of
seven in the second and in the
fourth. So as long as I travel
the sign with my term I can
rearrange these in any order
I like. So I'm just going to
rearrange the terms in the
order of XY, -X, and I'll go
ahead and do the (+)7Y first
and then the -7 term.
Just to keep my signs in
uniform going all the
way through.
So the next step in this
process is the actually group
those terms with common factors.
So I'll group my  XY and -X term
and then I'll group
my 7Y and my -7 term.
We'll continue this process by
factoring the GCF out of each of
the groups we've created. So
the GCF for first group is X.
and we divide that GCF out of
each of those terms we're left
with Y minus--
now I'm going to stop  here
because X divided by X is 1
and sometimes students won't
write that 1. They'll think that
it's just understood to be
there. It's only understood to
be there when it's
being multiplied,
in this case it's being
subtracted so we
actually need to write the 1
down there. Moving on to the
second group we have a common
factor of  (+)7 and
when I divide (+)7 out
of each of the terms I'm left
with Y and once again -7
divided by 7 gives me -1.
Now you should recognize that we
have a common binomial factor.
Well that common binomial factor
can be factored by dividing
it out of each of the terms and
bringing it out front.  And then
our next factor is made up of
what's left over.  So(Y-1)(X+7)
is the factored form of
the polynomial XY-7-X+7Y.
Let's check through
multiplication. So we'll simply
multiply our factored answer
by distributing each term
in the first binomial
with each term in the
second binomial.
So Y times X, putting it in
alphabetical order, gives us a
term of XY. And Y times
(+)7 gives  (+)7Y.
Moving onto distributing the -1.
-1 times X is -X
and-1 times 7 is -7.
Once again we can rearrange
these terms as long as we
travel the sign with them.
So let's see if we can match
that original polynomial
and rearrange into XY-7-X+7Y. 
And sure enough we end up with
the same polynomial that we
began with. So we are safe
to say that (Y-1)(X+7) is the
factored form of the polynomial
XY-7-X+7Y.
I hope this has been helpful
for you in learning how to
factor by grouping when you
have to rearrange the terms.
If you have any other
questions about this method,
please contact your
Highlands instructor.
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