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Welcome to Georgia Highlands
College  Math97 and Math99
video tutorials. in this video
segment we'll be answering
the question: how do you
factor a trinomial in
two variables in which the
leading coefficient is 1?
Well it's about the same
process that you take when
you're factoring a trinomial
whose leading coefficient
is 1 that's just
in one variable.
We follow the same steps;
set up your factors
Secondly you're going to ask
yourself what factors of C add
to make B. Thirdly, we'll fill
in those factors creating
our factored form, and then
we'll end up checking with 
multiplication. So let's
take a look at an example.
As you can see here I have
a trinomial that actually
has two variables in it.
One of the variables is X,
showing up in the form of
X^2 here and X here.   And I
have a Y and a Y^2.
I have two variables X and Y.
So if you think about it before
you ever get started on the
actual problem just like
we saw with X^2+BX+C;
we knew have to multiply
the first terms of
both binomial to make the X^2
term.
 Well it's going to have
the same idea with the Y^2 term
 on the end.   We know, not
including the 3 here, we'll deal
with that in a minute, but we
know that we had to multiply
Y by Y to make this Y^2 term on
the end. 
So to set up, to get ready to
factor this, I need to make sure
that I have a place for the sign
either positive or negative and
all so need to have a place for
the factors that make up the
coefficient of that last term so
I have spaces here and here.
From here on out it's the same
exact process that we talked
about with X^2+BX+C.
So I'm going to make my
factoring teepee again
and ask myself what are the
factors of 3, so we're going to
put our multiplication symbol
there to show we're going to
multiply up to make 3 and add to
make B which in this case is -4.
So we're looking for the
numbers that multiply to make
(+)3 that add to make -4.  So
if you think about that,
you're going have to have two
negative numbers because the
negative times a negative is a
positive, but a negative plus a
negative is a negative. So if
you think about it, the factors
of 3 are simply 1 and 3
but we need to take them in
their negative form.   So -3 and
-1.  -3 times -1 gives us (+)3.
 And -3 + -1 gives us -4.  So
we've found the
factors that we need to use to
fill in the blanks up here.
-3 and -1.  So we've got our
factored form of our trinomial
(X-3Y)(X- ,we don't write
1Y because it's understood
when it's multiplied
that the one is there.
We just write X minus Y. Now
let's check this just to be sure
So (X-3Y)(X-Y). Good old
distribution multiplying
each term in the first binomial
with each term in the second
binomial.  X times X yields X^2.
 X times -Y is -XY.
-3Y times X is -3XY, once again
we like to keep things in
alphabetical order. -3Y times -Y
is (+) 3Y^2.  
Combiningour like terms here
in the center we end up with
X^2-4XY+3Y^2. Which is the same
polynomial that we began with.
Therefore (X-3Y)(x-y) is the
factored form of the polynomial.
I hope that  this is helpful
for you to understand how to
factor a trinomial in two
variables with the leading
coefficient of 1.   If you have
any other questions regarding
this method please contact your
Highlands instructor.
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