Hi, guys.
Welcome to Algebra 1.
Today's lesson focuses
on factoring trinomials
with a leading
coefficient of 1.
I know it's a mouthful.
What you know
adding and subtracting
and multiplying positive
and negative numbers
will help you
get through this lesson.
You ready to get started?
Let's go.

(Describer) On a screen, x plus 3 times x plus 4.
She uses a stylus.

Okay.
To explain to you
what I mean by factoring,
I first want
to jump back to FOILing.
If I presented you with this
situation and asked you
to multiply these binomials
together using FOIL,
you'd probably write "FOIL"
somewhere
to keep it straight.

(Describer) She writes F-O-I-L.

Then you'd start
multiplying.
You'd say, all right,
first terms: X times X,
that's X squared.
I'm going to write that
underneath.
Outer terms,
so X times 4, that's 4X.
Inner terms,
so 3 times X, that's 3X.
Then your last terms,
3 times 4, that's 12.
Then you're at the point
to combine like terms.
Notice they're in the center
like they always are.
So 4X plus 3X, that's 7X.
You'd say, my answer
is X squared
plus 7X plus 12.
We use FOIL to figure out
what that product is.
Now let's think about
how we generated
these terms that we got.
We know that X squared,
we got that when we multiplied
the first terms together.
Right?
That was our first terms.
4X came about when we multiplied
the outer terms together.
Right?
That was our outer.
3X came about by multiplying
those inner terms together.
And the 12, from multiplying
the last terms together.
And then when we wrote
our final answer,
we ended up having to combine
the outer and inner term--
or the answer from multiplying
the outer and inner terms.
The first term
was still the first term.
The second term
was from combining
the outer with the inner,
and then 12 was our last term.
This was our work,
and this was our final answer.
Now, what about this?
What if I gave you this?

(Describer) She erases the work.

Get rid of all that.
Even got rid of this.

(Describer) She deletes the original problem.

Get rid of that.
There we go.
What if I gave you this?
What if I gave you this
trinomial?
This X squared
plus 7X plus 12?
And I asked you,
what were my factors?
What did I multiply together
to get this trinomial?
That is what it means
to factor a trinomial,
and in this case,
the leading coefficient is 1,
because the coefficient
of this first term is 1.
I'll show how we work our way
through this problem
and factor a trinomial.
I'm going to use
a different example.
I'm going to use X squared
plus 7X plus 10
to figure where--
or how we can factor this.
Now, to start with,
we know that the answer's
going to take that format.
I'm going to have
two sets of parentheses,
and I know that there's going
to be an X in this one
and an X in that one, okay?
If you forget,
let's flip back.
I erased it, but remember
this started as this.

(Describer) X plus 3 times x plus 4.

It always takes
that type of format,
where the X's are first,
right?
Flip back.
What I need
to figure out now
is what terms will
finish out these parentheses.
Okay?
I need to figure out
what two numbers will multiply
together to give me 10,
but when I combine them
I get 7.
I'll flip back to show you
what I mean by that.
Remember working this problem
out when it started like this?
We said our last term came
from multiplying 3 times 4.
And this inner term came about
when we multiplied--
let's get this work back,
that X squared plus 4X
plus 3X plus 12.
That was where
this all began.
That last term, like we said,
came from multiplying
3 and 4 together.
And that 7X came from adding
the outer and the inner.
So when I'm trying
to solve the next problem,
I need two numbers that
will multiply together
to give me the last term,
but when I combine them,
I get that coefficient
of that middle term.
Stay with me, I know
it's a little funky at first.
Let's start by listing
the factors of 10.
I need to figure out
what numbers multiply
to give me 10 at first.
I'll come over here
and factor 10.
1 and 10; 2 and 5.
And that's it for 10.
I've handled knowing
what numbers
have a product of 10.
Now I need to decide,
out of these pairs of factors,
how could I combine
either pair to get 7?
This is what I mean--
it's guess and check.
If I combine 1 and 10--
1 plus 10, that's 11.
So that's not going
to give me 7.
But 2 plus 5,
that will give me 7.
That means to finish out
my factors up here,
I would need a positive 2
and a positive 5.
I have factored
this trinomial completely.
If I were to multiply
this back out,
I would get this
as my answer.
I'll show you
to prove my point.

(Describer) X plus 2 times 2 plus 5.

Those are the factors
of that trinomial,
so to check your answer,
you could
multiply that out.
So FOIL: X times X...
that's X squared.
X times 5, that's 5X.
2 times X, that's 2X.
2 times 5, that's 10.
Then you're at the point where
you combine like terms;
you have like terms
in the middle there.
5X plus 2X
would give you 7X.
So X squared
plus 7X plus 10.
And that is the problem
that we started with.
That means we chose
our factors correctly.
Okay?
Let's try another one.
Factoring is something
you have to do a few times
to get the hang of it.
I might make it look easy,
but I've been
doing math a long time.
When I first learned,
it was frustrating.
Keep practicing and know
your multiplication facts,
and that will take you
a long way with these.
Let's handle this one.

(female narrator)
X squared plus 5X plus 6.

(Describer) X-squared plus 5x plus 6.

(instructor)
You got to factor it.
We know at least
it's going to take that format:
I'll have two sets of
parentheses.
The first term
is going to be X--
leading coefficient
of 1 in this problem,
so I'll write X and X.
Now I need to figure out
what two numbers multiply
together to give me 6,
but when I combine them,
I get 5.
Let's start out by getting
the factors of 6.
So I know 1 times 6 is 6.
And I know
2 times 3 is 6.
And those are all
your factors for 6.
With the factors of 6,
or the pairs of numbers
that multiply to give you 6,
you figure out which pair
combines to get 5.
1 plus 6, that's 7,
so that's not it.
Now, 2 plus 3, that is 5.
Those are the factors
that you need.
Back up in my problem,
that means I would need
X plus 2, X plus 3.
And you're all done,
you factored it completely.
And to check your answer,
remember you could
FOIL that out
and make sure
you were right.
Let's try some more.
All right,
I need to factor completely:
X squared plus 6X plus 8.
So I know I have this
to start with;
I know my answer's
going to take that format.
It'll have
X and an X out front.
Now I need to figure out
what numbers would multiply
to give me 8,
but when I combine them
I get 6.
Let's start by listing
the factors of 8.
1 and 8 will give you 8,
as would 2 and 4
if you multiplied them.
And that's it for 8.
Now, out of these pairs
of factors,
which pair could you combine
and get 6?
Well, 1 plus 8, that's 9,
so that's not it.
But 2 plus 4, that is 6.
So that means,
to finish out my problem...
X plus 2,
X plus 4,
and you're all done.
Let's try another one.
So I need to factor
completely.
X squared plus 7X plus 12.
So to start out,
I know my problem's
going to take that format,
with X's out front, okay?
I need to figure out
what numbers could I multiply
together and get 12,
but when I combine them,
I get 7, okay?
Let's start by listing
the factors of 12.
I know 1 times 12
will give me me 12,
and so would 2 times 6,
and so would 3 times 4.
Those are my factors for 12.
Now, which pairs
of these factors
would combine to give you 7?
That's the next thing
to figure out.
Well, 1 plus 12, that's 13.
2 plus 6, that's 8,
and 3 plus 4, that's 7,
so that's it.
We need X plus 3, X plus 4.
Okay? All right.
It is your turn.
Go ahead and press pause,
take a few minutes
and work your way
through these problems.
To compare answers with me,
press play.

(Describer) Title: Factor completely.
Number One: x-squared plus 9x plus 20.
Number Two: x-squared plus 6x plus 9.

(female narrator)
Factor completely.
Number 1:
X squared plus 9X plus 20.
Number 2:
X squared plus 6X plus 9.
Okay, ready to check?
Let's check.
All right.
For the first one,
X squared plus 9X plus 20.
That factored to X plus 5
times X plus 4.
And the second one,
X squared plus 6X plus 9,
that factored to X plus 3
times X plus 3, okay?
To see how I got those,
this is what I did.
First--
and now I need the pen--
I knew that my answer
would take this format.
I put
the X's in there also.
Now I needed to figure out
what numbers would multiply
together to give me 20,
but when I combined them
I get 9.
Come off to the side.
I know that 1 times 20 is 20,
so is 2 times 10,
and so is 4 times 5.
Now I need to see which of these
would combine to give me 9.
1 plus 20 is 21.
2 plus 10 is 12.
4 plus 5, that is 9.
So X plus 4, X plus 5.
And I was all done
with that one.
And remember, to check--
I will show you--remember,
to check, you can multiply
your products together
to get the trinomial
you started with.
I'll show you the check
to make sure
your answers are right
before you check your answers
against mine, before you
turn in your work in class.
Always check your work first.
If you multiplied that out,
that X plus 4 times X plus 5...
So FOIL, right?
So X times X.
That's X squared.
Outer:
X times 5, that's 5X.
Inner:
4 times X, that's 4X.
The last: 4 times 5.
That's 20.
Then you see you've got
like terms to combine.
The 5X and the 4X
will give you 9X.
Then you've got your X squared
plus 9X plus 20.
If you multiplied
your factors out,
you could check that
you got your answer right.
Let me show you
how I got number two.
First, remember,
always do this first,
write my parentheses
in there.
And there's an X
and an X first in each set.
Now I have to figure out
what numbers
would multiply to give me 9,
but combine to give me 6.
I know 1 times 9,
and I know 3 times 3.
Those are factors for 9.
Now I have to figure out
which pair of factors combines
to give me 6.
So 1 plus 9, that's 10.
And 3 plus 3, that is 6.
So that's how I knew
I needed X plus 3,
and X plus 3, okay?

(Describer) The next problem: x-squared minus 8x plus 15.

(female narrator)
The next problem: X squared
minus 8X plus 15.
All right, so, here...
you probably noticed that
I have a negative sign here.
Your process is the same.
I'll set up what my answer
will look like,
I'll get my factors,
etcetera.
But the negative puts
a spin on things.
Now you need to consider
you're working with integers,
so you could have positive
and negative numbers.
I wanted to start
with positive numbers
so you could get
the hang of it.
Now, as you've got
a hang of the process,
it's time
to step it up a notch.
I'm going to start out
doing this.
I know the answer's
going to take that format,
with X's first.
Now I can figure out
my actual numbers.
I need to figure out
what factors,
or what pairs of numbers
would multiply together
to give me 15,
but would combine
to get negative 8.
For a second
just ignore the signs.
Let's focus on factoring--
whoo,
that was a tongue-twister,
"focus on factoring" 15.
So factors of 15:
1 and 15,
3 and 5, right?
Okay, so here we go.
If you multiply
a positive number
times a positive number,
that's positive, right?
But if you multiply
a negative number
times a negative number,
that's also positive, right?
That's where those integer
operations come in place.
What I mean is, for example,
1 times 15 is 15.
Negative 1 times negative 15
is also 15.
So when you multiply
numbers together
that have the same sign,
your answer's always positive.
Throwback to pre-algebra
on that one.
Here, what you have
to think about
is that your product
is a positive 15,
but the sum that you're
reaching for is negative 8.
So the only way you can
multiply numbers together
and take the same two numbers
and get a negative answer
is if the numbers themselves
were negative.
So I have to consider
negative 1 and negative 15,
and negative 3 and negative 5,
because I know there's no way
I can combine a 3 and a 5
and get a negative number.
I can't combine
a positive 1 and positive 15
and get something negative.
I have to find
the negative cases.
For example:
negative 1 plus negative 15,
that's negative 16--
that's not it.
Now, negative 3
plus negative 5,
that is negative 8.
That's the sum
I was looking for.
What that means is,
over here in my factors,
I had X minus 3,
and X minus 5.
These ones aren't positive;
these factors aren't positive.
When you see negative signs,
you have to start considering
negative numbers also
when doing
your guess and check.
Let's try another one
with those negatives.
So X squared
minus 9X plus 20.
To start, you'll always get
those parentheses up there,
have an X in each group.
Now I need factors of 20
that would also combine
to give me negative 9.
We can already start to think
about this when we start.
If we multiplied
two numbers together
and our answer was positive,
the only way that combining
those numbers
would give us
a negative number
was if the numbers
were negative.
Let's get it on the board
so you can see.
So I'm factoring 20,
so know 1 times 20 is 20.
2 times 10 is 20,
4 times 5 is 20.
And that's it for 20.
Now looking at these numbers
as all positive,
there's no way I could combine
them and get negative numbers.
1 plus 20 will never equal
a negative number.
2 plus 10 will never
equal a negative number.
4 plus 5 will never equal
a negative number.
I'm trying
to get negative 9.
So the only way that
this could work out
is if both of my factors
were negative.
Because multiplying
these numbers together
would still give me
positive 20,
but combining them, I would
get something that's negative.
So let's see.
Negative 1 plus negative 20,
that's negative 21.
That's not it.
Negative 2 plus negative 10,
that's negative 12.
So that's not it.
Negative 4 plus negative 5,
that's negative 9.
So this is the pair
that I want.
So I need minus 4,
minus 5.
And that would be
my factors.
Let's try some more.
Remember, I said these,
you have to do a few.
The more you do, the more
comfortable you'll feel,
and it'll become
second nature.
Let's start out with, what?
Right, putting
those parentheses in there.
And putting the X in there.
Now I need numbers
that will multiply together
to give me 4,
but when I combine them,
I get negative 5.
Let's list the factors of 4.
I know 1 times 4 is 4,
and I know 2 times 2 is 4.
Now, combining
two positive numbers
will never give me
a negative number.
I know for this situation,
I need to consider
negative numbers.
Multiplying two negative numbers
together is still positive,
and combining two negative
numbers is a negative number.
So I need to figure out
what's going on here.
So negative 1 plus negative 4,
that is negative 5;
it was the first set
this time.
I'll show you why it's not
the second pair.
Negative 2 plus negative 2
is negative 4.
We wanted negative 5,
so it's not that one.
So for my factors,
I'd have X minus 1
and X minus 4.
Okay?
Let's try another one;
this one is different.
We have two negative signs
this time.
But same process.

(Describer) X-squared minus 2x minus 8.

Start out
with your parentheses.
You know there's going
to be an X in there.
For a minute ignore the sign,
and let's just list
our factors for 8.
So 1 times 8 is 8,
and 2 times 4 is 8.
But I need a negative 8,
so now think back
to multiplication of integers.
The only way to get
a negative number
for your answer is if
you multiply a negative number
times a positive number.
This is what I mean.
1 times negative 8
is negative 8.
Negative 1 times 8
is negative 8.
The only way
to get a negative number
for your answer
is if one of your numbers
is negative
when you're multiplying.
In this situation,
because I was trying to get
a negative 8,
one of these factors
is going to be negative.
We have to figure out
which factors should it be
so when you combine those
you also get negative 2.
You kind of have to do
a little guess and check.
Use your calculator
to avoid mental math.
Is there any way
you can combine 1 and 8
and get negative 2?
Mm-mm, because 1 plus
negative 8, that's negative 7.
Negative 1 plus 8,
that's positive 7.
There's no way I can get
negative 2 when the numbers
are a 1 and an 8;
that's not it.
Let's consider
the 2 and the 4.
Is there any way
to combine 2 and 4,
with one of those
being negative,
and your answer
be negative 2?
Yes, there is.
If the 4 was negative,
because 2 plus negative 4
is negative 2.
And that's what
we wanted for our sum.
That means for my factors,
I'd have X plus 2,
and X minus 4.
Actually, I have a positive
and a negative.
Let's do another.

(female narrator)
X squared plus 2X minus 15.

(Describer) X-squared plus 2x minus 15.

(instructor)
I'll start by writing out
the format of my answer.
I'm going to ignore
the sign for a second,
and I know
I need to factor 15.
Let's get the factors of 15.
So 1 and 15; and 3 and 5.
Now is when we consider
the signs.
We know we have
a negative 15.
So when we're multiplying,
if you get a negative number
for your answer,
you multiplied a positive
and a negative together.
As far as factors go,
one of these numbers
needs to be negative
in the pair.
I need to find which number
should be negative.
Let's think our way
through this.
I need 2X.
Is there any way
I could combine
a 1 and a 15 and get 2?
If either one of them
was negative?
If the 1 was negative,
I'd have negative 1 plus 15,
and that's 14.
So that wouldn't
give it to me.
And If the 15 was negative,
then I'd have 1
plus negative 15,
and that's negative 14.
That's not
what I need either.
It's got to be this pair,
but which one of these numbers
should be negative?
What if the 3 were negative?
Then I'd have negative 3
plus 5, which is 2.
That's what I need;
I need the 3 to be negative.
So I need X minus 3,
and X plus 5.
And those are the factors
that I needed.
You can always multiply
your factors together
to make sure what you got
for your answer is correct.
Okay?
It is your turn to try.
Take a few minutes
and factor these problems,
and when you're ready to check
your answer, press play.
Remember, you can check
your answer before that.

(female narrator)
Factor completely.

(Describer) Title: Factor completely.
Number One: x-squared minus 7x plus 12.
Number Two: x-squared plus 3x minus 4.
Number Three: x-squared minus x minus 6.

Number 1:
X squared minus 7X plus 12.
Number 2:
X squared plus 3X minus 4.
Number 3:
X squared minus X minus 6.
All right,
let's see how you did.
I'll factor that first one.
My factors were X minus 3,
and X minus 4.
When I factored
the second one,
my factors were X plus 4,
and X minus 1,
and when I factored
the third one,
my answers were X minus 3,
and X plus 2.
I'll show you
how I got those.
First thing I did,
I wrote down--
I need my pen--
I wrote down the format
of my answers,
or the format of my answer.
I knew I needed
to find numbers
that would multiply
to give me 12,
but combine to give me
negative 7.
I started out
by factoring 12.
I know 1 and 12
would give me 12, multiplying.
2 and 6 would,
and so would 3 and 4.
I know I need a positive 12
for the product.
So either both
of these numbers are positive,
or both are negative,
to figure out
which pair I need.
That's the only way
when you're multiplying
to get a positive number:
multiplying two positive
or two negative numbers.
Since the sum I need
is a negative 7,
a negative number,
it tells me these were
both negative numbers.
Because the only way
that the product is positive
but the sum is negative
is if the factors
were negative.
Let's see which pair
it is that we need.
If it were this first pair:
negative 1
plus negative 12,
that's negative 13;
that's not it.
Negative 2 plus negative 6,
that's negative 8.
So that's not it.
Negative 3 plus negative 4,
that is negative 7.
That means my factors
were X minus 3, and X minus 4.
That's how I got
the first one, okay?
Next, same process.
Start out by setting up
the format.
For this one,
I need to factor 4.
So I know 1 times 4
gives me 4,
and so does 2 times 2.
Now I consider the signs.
It's a negative 4 that I need.
So I know that
when I'm multiplying,
I get a negative number
for my answer
if I multiply a positive number
times a negative number.
In this case, my factors
will have
two different signs.
I need to find which one
of these numbers is negative,
because when I combine
the numbers,
I need to get a positive 3.
Play around
and see what happens.
With that first set,
if the 1 was negative
and the 4 was positive,
negative 1 plus 4, that's 3;
that was quick.
So that's the pair
that you need.
So X minus 1,
and X plus 4.
Okay? You're all done
with that one.
The last one,
I started out
by setting up the answer.
Okay?
So I know I need to factor 6.
So 1 and 6; 2 and 3.
Now, in this case,
it's a negative 6.
So I know when
I'm finding a product,
the only way my product
is negative is if my factors
were a positive number
and a negative number.
So one of these numbers
will be negative in my pair.
I have to figure out
which number is it.
You're trying to combine
and get a negative 1--
there's a 1 there
you don't see.
Play around to see
which pair makes negative 1.
So with 1 and 6,
if the 1 was negative,
negative 1 plus 6, that's 5.
So that's not it.
Now, what if the 6
were negative?
1 plus negative 6,
that's negative 5,
so that's not it either.
For the second pair,
if the 2 were negative,
negative 2 plus 3,
that's a positive 1,
but we need a negative 1.
So it's not the 2
that should be negative.
It's actually the 3.
2 plus negative 3
is negative 1.

That means your factors:
(X plus 2), (X minus 3), okay?
All right, guys,
hope you're feeling confident
factoring trinomials where
the leading coefficient is 1,
and I hope you saw how important
multiplication facts
and your knowledge
of integer operations
really help you
get through this lesson.
Hope to see you
back here soon. Bye!

(Describer) Accessibility provided by the US Department of Education.