Hi guys,
welcome to Algebra 1.
Today's lesson focuses
on factoring binomials.
Now you're going
to have to think back
and use your knowledge
of greatest common factor
to get through this lesson.
You ready? Let's go.
Let me take you back a bit
so you can remember
what it meant to be
the greatest common factor,
how to find
the greatest common factor.
It's all about
knowing which numbers--
in this case 15, 18.
I need to find out
what are the factors of 18?
What numbers can I multiply
together to get 18--
I'm sorry, to get 15--and then
do the same thing for 18.
What numbers can I multiply
together to get 18?
And then I just
look for the largest number
that they have in common.
Let me show you what I mean.
So 15, I'm gonna come...
right under here.
The factors of 15--
What can I multiply together
to get 15?
You have to think back
to your multiplication facts.
I know I can use 1 and 15.
1 times 15
will give me 15.
15 is odd,
so I can't use 2
to multiply by anything
to get anything 15,
but I can use 3.
3 times 5 is 15.
As far as 15 goes,
that's it for its factors.
Now let's look at 18;
we need to factor 18.
What pairs of numbers could you
multiply together to get 18?
Well, 1 times 18,
definitely--
that's always
a pair of factors.
One times anything
always gives whatever number
you start it with.
It's even,
so I know 2 would work.
2 times 9,
that gives me 18.
Will 3 times anything
give you 18?
It will.
3 times 6 will give you 18.
As far as 18 goes,
that's it.
If you don't feel confident
about your multiplication facts
and you think
you're leaving one out,
just use your calculator
and start trying things out.
Or you can go back
and get a multiplication table;
you may have used that
back in elementary school.
Let that help you
get your facts stronger.
Eventually
you won't need them at all,
and you'll just remember
what they are.
Okay, so look back here.
Since I've listed
the factors of 15
and the factors of 18,
I look for the largest number
they have in common--
their greatest common factor.
If you scan both sets,
you'll notice it's 3.
The greatest common factor--
often abbreviated as GCF--
for these two terms,
it's 3.
That's the biggest factor
they have in common.
Okay?
Let's try another one.
30x and 10.
You saw the x--
maybe it threw you,
but just ignore the x
for a second.
Let's focus on the number part
of this for a minute.
Factors of 30,
because I'm ignoring
the x for a minute.
I know that 1 times 30
will give me 30.
It's even,
so you start thinking about 2.
2 times 15,
that'll give you 30.
Okay? 3 times 10.
There's actually
one more pair--
5 and 6.
5 times 6
will also give you 30.
Then you come over here.
Actually, let's bring
that x back now.
There's an x here;
I'll teach you a little trick.
I'm just going to write
1x is involved over here
so that I know
there's a factor of x
that's included
in that 30x term.
That'll just help me
keep that in my head.
For the 10, I know 1 times 10,
that'll give me 10.
2 times 5,
that'll give me 10.
That's about it
as far as getting 10.
Now we're at the point
that we can scan our list
and see what's the greatest
common factor here.
I see they both have
a 1 in common,
and they both have a 5
in common,
and they both have
a 10 in common.
10 as far as my number
is concerned--
that's the greatest
common factor so far.
Remember I said
ignore the x for a second
and we'd come back to it?
The 30x term, there's an x
involved there.
X is a factor of that term,
but this is
just a plain old 10;
it's just a constant term.
There's no variable
involved.
As far as this problem
is concerned,
the greatest common factor
is just 10.
I can't write 10x
because that 10 didn't have
an x attached to it.
All right,
let's keep going.
I've got 22x squared
and 11x,
need the greatest
common factor.
I'm going to ignore
the x squared for a second
and just focus
on that 22 to get started.
I know 1 times 22
would give me 22.
It's even,
so 2 times something;
in this case,
11 will give me 22.
As far the 22 part
is concerned,
those are the only ways
to get 22.
Now let's look at the variable
part of this--that x squared.
I know that x squared means
that I multiply x times x
together to get x squared,
so I can say
this also has an x times x
as a pair of factors.
That 22x squared
includes a pair of "x"s
as its factors,
that x and that x.
So keep that in mind,
move over here to the 11x.
11 is a prime number,
so its only factors
are 1 and itself.
So 1 times 11,
that's the only way
you can get 11
when multiplying whole numbers.
I see that I have an x here,
and there's just one x;
I don't have
an x squared term.
Like we did
in that problem a second ago,
I'll write there's a factor
of x involved in this.
I'm going scan my list and see
what they have in common.
They do have a 1 in common,
but that's
not a big number here.
The biggest number
they have in common, 11.
As far as the number part
is concerned,
11 is the greatest
common factor
as far as
the coefficients go.
I have two "x"s over here,
x times x,
and there's just one x
for this term.
The greatest number of "x"s
that these terms have in common
is just one,
because both of these terms
have at least one x.
For this problem,
I'd say the greatest
common factor is 11x.
11 was the largest coefficient
they had in common,
the largest regular number,
and they had at least one x--
both of them
had at least one--
so I can write 11x,
greatest common factor.
Let's keep moving.
Now, greatest common factor
is going to help you know
how you factor binomials--
when you're adding together
two monomials.
The first step is to figure out
the greatest common factor.
I wanted make sure that
you remembered how to do that.
After you've got it,
divide each term
by your greatest common factor.
Okay?
Let me show you what I mean.

(female narrator)
Example 1: Factor completely.
15x squared minus 50x.
Okay, here we go.
"Factor completely."
When you see directions
like that,
all that's telling you to do
is factor this problem
in the simplest way,
so that when you've written
your answer,
there's nothing else
you could do to simplify it.
You want it
in its simplest form.
What I want to do here
is first identify
the greatest common factor.
I'm going to split
these terms apart
and think about them
separately for a second
to help me figure out
the greatest common factor.
I'm going to do
a little scratch work over here
and this is to get the GCF.
Right now, I'm focusing
on 15x squared.
I know there's a minus here
to keep in mind,
but to find
the greatest common factor,
I'll ignore it for a minute
and just say,
"I've got 50x as the term
I want to factor."
We start breaking this down.
For 15x squared think about
how you can get 15,
factors of 15.
You have 1 and 15, 3 and 5--
we handled 15 before
so you probably remember
those terms, those factors.
You have x squared
part of this,
so that means you have
a product of two "x"s,
so x times x--
that's also going
to be a pair factors for you.
Done with that first term,
so now let's handle
the second one.
Let's first handle the 50
and ignore the x for a second.
You know 1 times 50,
2 times 25,
you think about quarters.
3 times anything,
4 times anything,
but 5 times 10.
That pretty much sums up
your factors for 50.
Move that out of the way.
There's one x here,
so I'll write down
that I have one factor of x.
Now that I've got my list,
I'll pull out
my greatest common factor.
I scan the numbers first
and I see 5
is the greatest common factor
as far as the numbers go.
The 15x squared has two "x"s,
the 50x just has one.
So they have at least one x,
both of them.
So that means
my greatest common factor
between these two terms
is 5x.
All right?
We've completed step one.
We know what the greatest common
factor is for these terms.
Step 2 is to divide
each of these terms
by your greatest
common factor
to get the rest
of your answer.
I'll set up what format your
answer is going to look like.
You're going to have
your 5x
and there'll be a set
of parentheses right here.
We're trying to factor this,
or figure out what we'll
multiply together
to get this binomial.
We multiply 5x
times something,
and we need to figure out
that something.
That minus sign,
so I don't forget about it,
I'm going to bring it
in my parentheses,
so I remember that
is separated by subtraction.
Now that I've got that,
get us
some more room to work.
Let's get this out of the way.
We found the greatest
common factor.
Let's go ahead
and handle step two,
which is to divide each term
in your binomial
by your greatest
common factor.
That's how you figure out
what goes in your parentheses.
Okay? Let's start.
You've got that
15x squared.
We're going to divide it
by our greatest common factor,
that 5x.
Now you got to think back
to those rules of exponents,
remember how you handle
situation like this.
You divide the coefficients,
so 15 divided by 5,
that's 3.
Remember when you don't see
an exponent, there's a 1--
invisible, okay?
Because to divide
that x squared by x,
remember you need
to subtract those exponents,
that rule of exponents.
2 minus 1, that's just 1.
So this is 3x
to the first,
or I could just write 3x,
which is what I'm going to do,
write 3x.
I know that this first part
of what goes
in my parentheses is 3x,
which is what I get
when I divide that first term
by the greatest common factor.
Now I'm going to move on
and divide 50x
by the greatest common factor,
and that's how I figure out
what goes in this empty space,
what term goes here.
So now 50x divided by 5x--
still remembering
our laws of exponents,
divide the coefficients.
So 50 divided by 5,
that's 10.
Remember, there's some
invisible ones
that I'm not seeing
right there.
So x to the first
divided by x to the first--
remember, when you're dividing
you subtract your exponents;
so 1 minus 1, that's zero.
I am going to write that down
so you know what I mean.
It's x to the zero power.
Remember, anything we raise
to the zero power is just 1.
This is essentially saying
that my answer is 10 times 1
because that's what I get
when I handle the "x"s,
but just write that as 10--
10 times 1 is just 10.
And 10 is the number
that's going to go right here.
When I factor completely,
15x squared minus 50x,
you get 5x times 3x minus 10.
What that means
is if you multiply this out,
using the distributive property,
this would be your answer.
Factoring
reverses multiplication.
It's like you're answering
the question,
"What did I multiply together
to get that first problem?"
That's basically
what we're doing every time,
trying to figure out
what do we multiply together
to get whatever
they're asking us about.
Let's try another one
to make sure you got it.
"Factor completely.
3x plus 15."
It looks a little different
but the steps are the same.
It's just a binomial;
you're just going to factor it.
First you know you need to find
your greatest common factor.
I'm going to write that
over here,
that I'm finding
the greatest common factor
between 3x and 15.
As far as the 3x
is concerned,
the coefficient--
3 is prime.
So we know the only factors
are 1 and 3.
There is that one little x
hanging out with that 3,
so I know
I have a factor of x,
and I'm done
as far as 3x concerned.
Now the 15,
we've handled 15 a few times.
We're trying to give you numbers
that are small and familiar
so that you start to feel
confident about your factors.
So 15: 1 and 15, 3 and 5.
And it's just
a constant term;
there's no x,
no variable attached.
I don't have any "x"s
to write as factors of 15.
I'm at the point now
where I can find
the greatest common factor.
You look at your list.
All they have in common is 3.
The greatest common factor
between 3x and 15 is 3.
Step one, all done.
Remember, this is when,
on the last problem,
I set up the format
of my answer?
I knew I'd have 3 and then
something in parentheses.
This one's a plus,
3x plus 15,
so I'm going to put
the plus sign in between
just to keep
my signs straight.
Now we're ready for step two.
Go ahead and divide our terms
in the binomial
by the greatest common factor.
Okay.
We need to divide
3x by 3.
3x divided by 3.
Remember, when you divide
the same number
on the top and bottom,
it just cancels out
because it just equals 1.
All you're left with that
first division problem is x.
The first thing that goes
into my parentheses,
the first term is x,
and now I need to divide 15
by the greatest common factor,
by 3.
So 15 divided by 3, that's 5.
And so 5 is the second term
that goes in your parentheses.
You're all done.
When you factor 3x plus 15
completely,
your answer
is 3 times x plus 5.
Remember, you can check that
by multiplying that out
and ensuring you get
what you started with.

(narrator)
Example 3: Factor completely.
10x squared plus 4xy.
All right, here we go.
This one's involving
more than one variable
because I know you saw the y,
but the process
is exactly the same.
We're going to find
the greatest common factor,
and then we're going to divide
each of these terms by it.
Let's get the greatest
common factor first.
I've got 10x squared,
and my other term is 4xy.
You think to yourself,
"Okay, she said
ignore the variable,
look at the coefficient,"
so ignore that x squared
and focus on the number first.
What are the factors of 10?
So 1 and 10,
2 and 5--
that's it for 10.
Here you have x squared,
so you remember
how you represented that?
That is the product
of x and x.
Okay?
All done for 10x squared.
Let's factor 4xy.
Ignore the variables;
just focus
on the 4 to start with.
You got 1 and 4,
that'll give you 4,
and 2 times 2
will give you 4.
Then you have
an x and a y,
so you have a product
of x and y.
That means you have two
separate variables, that's all.
Now we look for the greatest
common factor.
You scan the numbers
and the largest one
they have in common is 2.
They both have
at least an x,
but they don't each have a y,
so that's all
I can deal with, really.
My greatest common factor
is 2x for that problem.
Okay?
Step one is all done.
Now you're ready for step two.
You're going to take
your greatest common factor.
Let's set up
the answer over here.
I know I'll have 2x and
I'll have a set of parentheses,
and the sign in this one
is going to be a plus.
Now I'm ready to divide
each of these terms
by my greatest common factor;
let's see what I get.
Let's get this out of our way
and let's start dividing.
I need to divide
10x squared by 2x.
So 10x squared divided by 2x.
So, 10 divided by 2,
that's 5.
This is just 2x.
Remember, there's
an invisible 1 there.
So x squared divided by x,
you're really just subtracting
these exponents.
So 2 minus 1, that's 1.
So 5x to the first--
remember we don't
generally write when
it's a 1 as an exponent--
so 5x.
That's what will go up there.
Okay?
Now same thing.
Divide the next term
by your greatest common factor.
So 4xy divided by 2x.
Let's handle the numbers first,
the coefficients.
So 4 divided by 2 is 2.
There's the invisible 1 here,
an invisible 1 here.
Let's go ahead and put
the invisible 1 with the y too.
If I handle my "x"s,
x to the first divided by x
to the first,
you subtract your exponents.
So 1 minus 1, that's zero.
Really this is x
to the zero power.
Remember the 1
when we first started,
anything to the zero power
is 1.
Basically when you run
into this situation
and you're finding
your greatest common factor,
you can consider the "x"s
just wiped out,
because if the answer
ends up being 1,
you don't really need
to write "times 1."
It's not going
to change anything.
That 2 times 1 is just 2.
When I see that's 1, all right,
I'm done with the "x"s.
Let's look at this y.
I've got y to the first
and I don't have
any y's on the bottom,
so that means that that y
just keeps on hanging out.
It's 2y.
When I handle
this division problem,
the answer is 2y.
That's what's going
in my parentheses.
And you are all done.
When you factor
that binomial completely,
you get 2x
times 5x plus 2y.
Do you remember all along
that I've been saying that
if you wanted to check it,
you can multiply
what you got for your answer,
actually distribute this
throughout,
and make sure you get
what you started with?
I'm going to show you
what I mean now.
This is a way
that you can check your answers
and know if you're right
before you press play again
to see what I got.
You can do this
to check and see
if what you got
is the right answer.
I'm going to rewrite this.
I need the pen back first.
So, 2x and then 5x plus 2y.
This is what I mean by,
"You can just multiply it out
to check your answers."
Just use
the distributive property.
2x times 5x,
that's 10x squared,
and then 2x times 2y--
well, 2 times 2 is 4,
and x times y is just xy.
The answer we got
was the problem
that we started with.
You can always do that
to check your answer.
Like I said, factoring just
reverses multiplication.
If you multiply your factors,
you'll know if what you did
was right.
Okay?
It's your turn to try.
Go ahead and factor these
completely,
and if you want,
I encourage you
to check your answers
and see if what you got
is correct
before I reveal
what these answers are.
Press play when you're ready
to see what I did.

(narrator)
Factor completely:
Number 1:
3x squared minus 21x.
Number 2:
8y squared plus 6xy.
Number 3: 15x plus 9.
All right,
here's what I did.
For the first one,
my factors were 3x
and x minus 7.
3x times x minus 7.
For number two,
I got 2y times 4y plus 3x.
For number three,
I got 3 times 5x plus 3.
Let me show you
how I did it.
So, I followed our process.
First thing I did was find
our greatest common factor.
So, greatest common factor--
need the pen back.
Here we go.
I wrote down that I had
a 3x squared,
and I wrote down
I had a 21x,
and I just started factoring.
I ignored the x squared
for a second
and just focused on the 3.
So 3 is prime, so that's
an easy one to factor.
It's just 1 and 3,
and then you had x squared,
so you know that's the product
of an x and an x.
Okay?
That's it for 3x squared.
For 21x--21--
1 times 21 will give you 21
and so will 3 times 7.
You only have
one x over here.
So, one x as a factor.
Then I looked
for my greatest common factor.
They had a 3 in common--
that's the largest number.
And they each had
at least an x.
That meant my greatest
common factor was 3x.
Now that I knew that,
I set up my answer,
so I knew I'd have a 3x,
I knew I had
a parentheses,
and I see there should be
subtraction in between.
Let's get some more room
to work up here.
Now divide each of these terms
by your
greatest common factor,
exactly what I did.
3x squared divided by 3x--
so, 3 divided by 3,
that's just 1,
so basically,
these are cancelling out.
Then I have x squared
times x to the first,
so subtract these exponents--
2 minus 1, that's just 1.
So, it was just x
to the first, or just x.
Okay?
Then divide your 21x by 3x.
21x divided by 3x.
Handle the coefficients first.
21 divided by 3,
that's 7.
X to the first
divided by x to the first,
so I know I need to subtract
those exponents--
1 minus 1 is zero.
Remember anything
to the zero power is 1.
Multiplying 7 by 1
doesn't change anything
so I know in my mind,
"Okay, I've handled this one."
So it's 7.
That's how I got the first one,
3x times x minus 7.
In the second one,
did the same exact process.
My greatest common factor...
I factored 8y squared
and I factored 6xy.
My factors of 8--
1 and 8, 2 and 4,
and that's it for 8.
For the y squared part,
you know that's the product
of 2 "y"s.
You have a y
and a y as factors.
For the 6,
you think of factors of 6.
1 and 6, 2 and 3,
and here you have
an x and a y.
So x times y--
that's going to be a factor
of that term.
You scan: What's your
greatest common factor here?
The largest number
they have in common is 2
and they each have
at least one y.
Your greatest common factor
here, 2y.
Now I'm going to go over here
and set up my answer.
I know I'll have a 2y,
I know I'll have
parentheses,
and I know there's going to be
a plus sign in between.
Let's get some more room
to work up here.
Eraser, here we go.
All righty.
And I need to divide
each of these terms
by the greatest common factor.
So 8y squared divided by 2y.
So, 8 divided by 2,
that's 4.
Y squared divided by y
to the first,
subtracting my exponents--
so that's just y.
I have 4y right here.
And then now
6xy divided by 2y--
coefficients first,
handle those--
so 6 divided by 2,
that's 3.
I have an x up top,
but none on the bottom,
so x just carries along
for the ride.
Then I have y to the first
divided by y to the first.
1 minus 1,
that's zero.
Anything to the zero power
is 1,
so basically
it never happened,
because multiplying
3x times 1
isn't going
to change anything.
Over here, I have 3x.
That's how I made my way
to the answer on that one.
On the last one,
same process.
Greatest common factor.
I'm factoring 15x,
and I'm factoring 9.
You're probably a pro
at factoring 15 at this point.
So 1 and 15, 3 and 5.
And you know you have
one factor of x.
You're done with 15x.
Nine--
1 times 9 will give you 9,
and so will 3 times 3.
And it's a constant term
so I don't have
any variable factors.
I look and I find
my greatest common factor.
They each have a 3 in common,
and that's it.
So 3 is the greatest
common factor,
so now I'm done
with that scratch work.
Come over here
and set up my answer.
I know I have a 3,
I know I have
a set of parentheses,
and I know I will have
addition in between, right?
Let's get
some work space here
and get that out of the way.
All right.
Now I need to divide
each of these terms by that 3,
by my greatest common factor.
So 15x divided by 3--
so 15 divided by 3,
that's 5.
X--and I don't have any "x"s
on the bottom
so that's just
carrying along, so 5x.
And the 9
divided by 3 is 3.
That second term is 3.
All right.
I hope you're feeling confident
about factoring binomials
and I hope
you saw how important
knowing the greatest common
factor was in these problems.
Hope to see you soon.
Bye, guys.