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This video is going to be about factoring a polynomial,
and we're going to do this by factoring out the greatest common factor.
So let's see what this means.
Here we've got a multiplication problem, I'm multiplying this monomial, 5x squared,
times the binomial 2x minus 3.
So basically I'm just distributing this 5x squared across the binomial.
So I've got times 2 is 10,
x squared times x is x to the third,
and then
5 times negative 3 is negative 15,
and since there's no x over here,
I just have the x squared,
and I get this result.
Now if I had done this backwards and started with this,
and tried to get to here,
I would be factoring this. In other words, when you factor something you're looking
for
two things that you can multiply together, they may be just numbers,
they may be polynomials, you're looking for two things you can multiply together
that will give you this solution. So In other words, I want to figure out - how did we get here?
The numbers that make this up are called the factors
of this polynomial,
or the polynomials
that make it up are called the factors of the polynomial.
So let's see how we got here.
Let's start with
10x to the third
minus
15x squared.
Now what I'm looking for when I'm looking for the factors is going to be some monomial
over here
and a polynomial
that I can fit in these parentheses.
Remember, I'm working backwards trying to here as if I don't know what this is.
So I've got a 10 and a negative 15,
and I know that I can divide them both by 5.
I'm gonna put a 5 over here,
and 5 into 10 goes 2 times,
and so that tells me that when I
multiply... In other words if I started here and wanted to get back to here...
5 times 2 is gonna give me the 10.
And 5 into negative 15
is negative 3.
Negative 3 times 5 is negative 15.
Now when you get to the variables,
it's really simple.
All you do
is look at
your variables - I've got two x's -
then look at the exponents.
Choose the smallest exponent
and that's going to be
what you can factor out. In other words, I'm factoring out an x squared.
And then figuring out what's left,
I take this x to the third,
divided it by x squared,
and I get an x.
I take this x squared,
divide it by x squared,
I get a 1.
I don't have to write a 1 here since negative 3 times 1 is still gonna be
negative 3.
And this will be my factorization
of this polynomial.
If you want to check to make sure your work is correct,
take your factorization
and multiply it out.
So 5x squared times 2x
is 10x to the third.
5x squared times negative 3 is
negative 15
x squared,
and that's where I started.
If you get back to where you started, then your work is correct.
Let's look at another one.
After you've done enough of these, they just
become kind of automatic.
I've got 6a to the third, b squared
plus 9a squared b,
and I want to find out
what monomial,
which I'm going to put here,
was multiplied
by what
polynomial,
which is going to go in the parentheses.
So looking at the coefficients first,
I've got a 6 and a 9,
and I realize
that both of them came about by being multiplied by 3.
So I'm going to factor out a 3.
Factoring a 3 out of the 6 is going to leave me with a 2,
and factoring a 3 out of 9,
will leave me with a
positive 3.
Then I go on to my variables. I've got a's and b's, so let's start with the a's.
a to the third and a squared.
Well, 2 is my smallest exponent,
so I'm going to factor out
a squared.
Factoring
a squared out of a to the third, is just dividing a to the third by a squared.
So I'm subtracting this 2 from this 3,
and I just get an a over here.
Dividing a squared by a squared
I just get a 1,
which I don't have to write because 3 times 1 is still gonna be 3.
I've now got a b squared
and a b.
Remember, any time you don't see an exponent, you can just put a 1 in to remind you that the exponent
is 1.
So the smallest exponent I have is 1.
That means I'm going to factor out a b...
I'm sorry I jumped ahead...
factoring a b out of b squared is going to give me a b.
Factoring a b out of b is going to give me a one, so once again I don't need that 1.
(Might as well move this parentheses a little closer.)
and this is going to be my answer.
Multiplying this back
to make sure I'm right...
3 times 2 is 6
a squared times a is a to the third,
b times b is b squared,
3 times 3 is 9,
and then I've got an a squared and a b,
and that's where I started,
so this works.
Here's one more.
In this case I've got a trinomial. I've got three terms.
This is no harder,
just a little more work involved.
So I'm looking for
a monomial
and then a trinomial that I'll put inside these parentheses.
So the first thing I see is everything is negative,
so I'm going to factor out
some negative
factor.
Looking at the 4 and the 8, I realize that I could divide them both by 4,
but I've got to be careful, because I can't divide 10 by 4.
So that means 4 is not going to be the greatest common factor for all of them.
So looking again, I realize that they're all even,
so I know I can factor out a 2.
Taking this negative 4 and dividing it by negative 2
is going to leave me with a 2 here.
The negative 8 divided by negative 2
will leave me with a positive 4.
Negative 10 divided by negative 2
will leave me with a positive 5.
Going on to the variables, I've got x to the fourth, x to the third, and x squared.
2 is my smallest exponent,
so I'm factoring out
x squared.
x to the fourth
divided by x squared will leave me with an x squared over here.
x to the third
divided by x squared
will leave me with an x.
And
x squared divided by x squared will leave me with just a 1.
Going on to the y's,
I've got a 2, a 3 and a 5 as my exponents. My smallest exponent is 2,
so I'm going to factor out a y squared.
y squared divided by y squared is just a 1,
so I don't have to write that there.
y to the third
divided by y squared is just y,
so I'll put a y in there.
And y to the fifth divided by y squared
will give me a y to the third.
So this is going to be the factorization.
And again,
to check this, we would just multiply it out and make sure we got back to where
we started.
So the basic processes is this...
you're looking at your polynomial
and what you want to do is break it down into
a monomial, which is going to be the greatest common factor
of this polynomial,
and then you want to find out what polynomial you have left that that
monomial multiplies.
So you start with your constants,
see what with the biggest number is you can divide into each of the
constants,
that's going to come out
as the coefficient
of your
monomial.
Whatever that is, in this case negative 2, you'll divide that
into each of the coefficients that youstarted with,
and that will give you the coefficients of your
polynomial in your answer.
Then you look at the exponents for each of the variables. You take the
smallest exponent
and that will let you know what the variable is in the monomial,
and then you'll factor that out of each of the exponents that you started with,
and use those in the polynomial that's your answer.
Okay,
so
this is not that hard. It's just a matter of practicing enough of them so they
become routine.
Take care.