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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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