Hey, guys.
Welcome to Algebra 1.
Today's lesson focuses
on finding special products.
Remember how you multiplied
binomials together
because that will
come in handy in this lesson.
You ready to get started?
Let's go.
Okay.
Let me take you back,
just to jog
your memory about
multiplying binomials
together.
Now, we used FOIL
to help us get through this.
It's one strategy;
you may know others.
But it helps us multiply
binomials together.
I see this problem
and I'd say,
F, for FOIL is telling me
to multiply
the first two terms together.
Let me move this up
to get some space.
Okay, so I'd say,
all right, X times X,
that's X squared.
I'd keep
working my way through FOIL.
So, the O,
those are my outer terms.
So X times negative 6,
that gives me negative 6X.
So then,
I'd come back up here,
I'd write my minus 6X.
Now, I'm at the I,
so I'm at my inner terms.
So 6 times X...
that's 6X,
so I have plus 6X.
Then, the L,
so I'm at my last terms.
So 6 times negative 6,
that's negative 36,
right?
So we'd bring that
up here.
After we're finished FOILing,
we'll look at our terms
and see if there were
like terms to combine,
and there are.
Okay, let's get this work
out of the way.
Okay.
Here, I can
combine this negative 6X
with this positive 6X.
Negative 6X plus 6X,
that's just 0.
That's nothing, so these,
when you combine them,
just completely wipe out.
And you're left with X squared
minus 36.
We used FOIL to
work through that one.
I'll do a few more.
You'll notice a pattern
whenever my factors are set up
in this particular way.
Look at these factors.
Look at this next one,
and you're going to start
to notice something.
All right, so this one,
I'm multiplying X plus 4,
times X minus 4, right?
I'm still using FOIL.
I won't show
my work on the bottom
while I'm FOILing through
since you've already
done FOILing
and you probably
already know the steps.
So for the F,
the first two terms,
so X times X--
I know that's X squared.
All right,
so my O, outer terms.
So X times negative 4,
that's negative 4X,
right?
Still with me?
Okay, the I,
so my inner terms--
so 4 times X,
so that's plus 4X.
Then, my L,
so my last terms,
so 4 times negative 4,
that's negative 16, right?
Now you're at the point
you can combine like terms.
You have some
in the middle here.
Combine that negative 4X
with that positive 4X.
So negative 4X plus 4X,
that's 0,
just completely wiped out.
Really, you're just left
with X squared minus 16.
Starting to notice
anything yet?
Let's do a couple more.
So now I've got--
let's move this up
a little bit--
X plus 7, times X minus 7.
So I'll bring FOIL up here,
keep me going.
So my first terms:
X times X,
that's X squared.
My outer terms,
so X times negative 7,
that's negative 7X,
so minus 7X.
My inner terms, 7 times X,
so that's 7X,
so plus 7X, okay?
Then, the Ls
are your last terms.
7 times negative 7,
that's minus 49, okay.
Remember,
now we're at the point
of combining
your like terms.
They're in the middle here.
So, negative 7X plus 7X,
that just wipes out, right?
That's 0.
You're left with X squared
minus 49.
Those wheels turning?
See something
that's going on here?
Let's look at it all together.
The first product,
when you multiplied X plus 6,
times X minus 6,
our answer was X squared
minus 36.
Then, we multiplied X plus 4,
times X minus 4;
our answer was X squared
minus 16.
And when we multiplied
X plus 7, times X minus 7,
our answer was X squared
minus 49.
What's happening
in this special product--
on the left,
my factors--
they're a sum
and a difference.
For each,
I have the sum of two terms
and then their difference.
The sum of two terms,
then their difference;
the sum,
and then the difference.
When we have a problem
that's written in the form
of M plus N,
times M minus N--
to make it general,
to use some variables--
but it's written
in that format:
a sum and then a difference,
your answer is always
in the same form.
Look at our answers
over here.
So for the first one,
let's identify
the M and the N.
In this one, our M is X...
and our N is 6, right?
Those are our M and our N.
Our answer
was X squared minus 6 squared,
which is 36.
Keep going with me.
Over here, our M was X--
I'll write the X here--
and our N was 4, right?
Those were our sum
and our difference.
We added and subtracted,
and our answer was X squared
minus 16,
which is 4 squared,
right?
Let's look at this last one.
In this case,
our M was X.
What was our N here?
Our N was 7.
And then our answer
was X squared minus 49,
which is 7 squared.
The format of our answers
is also always
taking the same shape.
It's always going to be
whatever your M is, squared,
minus whatever your N is,
squared.
That is the terms' sum
and difference.
Then, we call the answer
"the difference of squares."
Whatever those two terms are--
if written in this format,
when finding your product--
the answer is this format.
Square the first one,
square the second one--
focusing on those terms--
and it's the difference
of those two, okay?
Let's do a couple
to get the hang of it.
So, find this product.
It's in that format of a sum.
I have the sum of X and 9,
then the difference
of X and 9.
With this shortcut,
you don't need to do FOIL.
Recognizing that pattern,
just use what you know.
Because you know when things
take the shape
of M plus N,
times M minus N,
your answer is always
M squared minus N squared.
Let's figure out
what the M and the N are.
We can skip FOIL.
In this case,
I see that M is X,
and the N is 9, right?
Following this special
products rule over here,
I'll square these terms,
and it's the difference of them.
This answer would be X squared
minus 81,
which is 9 squared,
right?
Let's do another one.
Okay, so here
I see my first--
or my M is X; the N is 10
because it's in the format
of X plus 10, X minus 10.
I'll just focus
on the M and the N.
So, my M is X,
so I need to square that.
That's X squared.
Then, the N is 10,
so I need to square that.
So 10 squared, that's 100.
Then you're done.
You could avoid FOIL
completely with this.
That's the trick
of these special products.
All right,
so we're at the point
where you can try some.
I've got products
on the left
and some answers
on the right.
Press pause,
take a few minutes,
and see how you do
with these.
To compare answers with me,
press play.

(female describer)
Match.
On the left, number 1:
X plus 5, times x minus 5.
Number 2:
X plus 11, times X minus 11.
Number 3:
X plus 3, times X minus 3.
On the right: X squared
minus 9, X squared minus 25,
and X squared minus 121.

(instructor)
You ready to check?
All right, so for number 1,
I have X plus 5,
times X minus 5.
So, I think, okay,
I see my M is X.
That first term is X,
the second one's 5.
I'm looking for X squared
minus 5 squared,
and that is this one
right here:
X squared minus 25.
Okay?
Here, I see
my first term is X,
the second one's 11
for this sum
and difference here.
So I'm looking for X squared
minus 11 squared.
I scan,
I see it's that last one.
All right.
The last one here...
You could use
process of elimination,
but the math teacher in me
won't let you.
Let's think
our way through it.
That the first term is X.
Ignoring my signs,
I have X.
The second one's 3.
So I need X squared
minus 3 squared, and that's 9.
Here we go.
That's how I knew that
was that one.
All right?
Now, you could also,
if you're not feeling
comfortable
about recognizing
that pattern,
just use FOIL
for your answer, okay?
I'll show you
a different strategy.
To use FOIL
without recognizing
those special products
all the time, that's fine.
Or use it
to check your answer.
We already knew this one,
the answer should be X squared
minus 25.
To verify that,
to check that,
go ahead and FOIL it out
to check.
All right?
Your first terms,
X times X--
so I know that
that's X squared, all right?
The outer terms,
so X times negative 5,
that's negative 5X, right?
Then your inner terms,
5 times X,
so that's positive 5X.
Then L, so your last terms,
so 5 times negative 5.
So that is minus 25.
Look for where you can
combine your like terms.
We're noticing they're in the
middle, generally, with these.
I have negative 5X plus 5X,
that just wipes out; that's 0.
So you have X squared
minus 25.
That is another way
to verify your answer.
See if you're right
before pressing play
to see what I got.
Here,
if you wanted to check,
you could make sure
you were right.
The answer was
X squared minus 121.
Let's just verify we get that
when we FOIL this out, too.
Let's go ahead
and write FOIL.
Okay, so the F,
so X times X,
that's X squared.
The O, so the outer.
X times negative 11,
so that's negative 11X.
Then my I's,
or my inner terms--
11 times X,
so that's positive 11X.
Then, my last terms,
so 11 times negative 11,
negative 121.
You see
those two like terms
right in the middle,
we can combine these.
So negative 11X plus 11X
wipes out; it's just 0.
And you're left with X squared
minus 121, okay?
You checked it,
you verified it, it was right.
Let's check this last one.
The answer
was X squared minus 9,
so let's just verify that
with FOIL.
All right,
so our first terms--
so X times X, I know that
that's X squared, right?
Our outer terms,
so X times negative 3,
that's negative 3X, right?
Inner terms, so 3 times X,
that's 3X.
Then L, so our last terms,
3 times negative 3,
that's negative 9,
so minus 9.
You see you have
your like terms,
you can combine
right in here.
So negative 3X plus 3X
wipes out
and you're just left with
X squared minus 9.
So we verified our answer,
we're good to go.
All righty.
There's another type--
or two more types
of special products
that I want to show you.
Knowing what you do
about exponents,
we have
the quantity (X plus 3)
raised to the 2nd power.
When you raise something
to the 2nd power,
that means you're
multiplying it by itself.
This X plus 3
raised to the 2nd power
just means X plus 3
times X plus 3.
That's what that means to do.
To get this product,
what do we do?
We're going to FOIL, right.
Okay.
So our F, our first terms,
so X times X,
X squared.
Outer terms,
so X times 3, 3X.
Inner terms,
so 3 times X, is another 3X.
Then, our last terms,
so 3 times 3, that's 9.
You have some like terms
we can combine
in the middle.
So 3X plus 3X,
that's 6X, right?
So X squared plus 6X plus 9.
So when we raise X plus 3
to the 2nd power,
we get X squared
plus 6X plus 9, okay?
Let's do more.
There's another pattern here
to spot.
So, X plus 5 squared,
so you see that exponent--
what does it mean?
If it's to the 2nd power
then I'm multiplying X plus 5
by itself, all right?
So X plus 5, times X plus 5,
right?
Then how do I handle this?
I FOIL, right?
Okay, so the F,
my first terms,
X times X; X squared.
Now my outer terms,
so X times 5, that's 5X.
Now my inner terms,
so 5 times X, that's 5X.
Then, now my last terms,
so 5 times 5,
that's 25, okay?
After you FOIL,
what do you do next?
Right,
combine your like terms,
and you see you have two
in the center.
You've got 5X plus 5X,
that's 10X, right?
So X squared
plus 10X plus 25, okay?
When we raise X plus 5
to the 2nd power,
we end up with X squared
plus 10X plus 25, okay?
Starting to see
something yet?
Let's do a few more.
We're switching up
a bit by changing the sign,
but I still think you'll notice
what's going on here.
So we see that
we're raising
that X minus 4
to the 2nd power.
That means we're going
to multiply it by itself.
I've got my X minus 4,
times X minus 4.
FOIL it out, right?
Okay, so your first terms,
X times X,
so that's X squared,
okay?
Then, your outer terms,
so X times negative 4,
negative 4X.
Inner terms,
so negative 4 times X,
that's another negative 4X.
Then, the last terms,
so negative 4 times negative 4,
that's a positive 16.
Okay, so then, right,
combining like terms,
so negative 4X minus 4X,
that'll give me
negative 8X.
o X squared
minus 8X plus 16.
Raising X minus 4
to the 2nd power
gives me X squared
minus 8X plus 16, okay?
I'll do one more.
Let's see if you've noticed.
You've probably noticed by now,
like we've been doing,
that this means X minus 7,
in this case,
times X minus 7, right?
Then, let's FOIL.
Okay, so our first terms,
X times X;
X squared.
Our outer terms, so X times
negative 7; negative 7X.
Then our inner terms,
so negative 7 times X,
that's another negative 7X.
Then, negative 7 times
negative 7, so that's 49, okay?
Again, like terms to combine,
so negative 7X minus 7X--
you combine that,
you get X squared minus 14X
right in there,
plus 49.
Okay, so we raised
X minus 7 squared--
or raised X minus 7
to the 2nd power--
and got X squared
minus 14X plus 49.
Look at the answers
all together
and see if you spot it.
Look at the ones
that we were adding first.
When we multiplied,
when we wrote--
raised X plus 3
to the 2nd power,
we got X squared
plus 6X plus 9.
When we raised X plus 5
to the 2nd power,
we got X squared
plus 10X plus 25.
So thinking about
what we noticed
in that first special product,
consider these terms
separately:
that you have an X,
and then a 3,
in this first one.
You notice,
if I write this again
as M plus N squared here,
because I'm squaring
my two terms,
my M and my N--
that sum.
I consider the M first,
consider this first term first.
It's that
first term squared
for both of these.
You end up squaring that
first term for the answer.
I'm going to write
M squared here.
Look at this middle term.
Notice that as far as
the coefficient is concerned,
it's double that N,
because 3 times 2
would give you 6, right?
2 times 5 would give you
10 in this one.
That's how we're getting
the number,
but where did
the X come from?
Notice this;
I'm going to write this first.
You're actually
doubling the product
of your M and your N.
Think about that
for a second.
I've got X and 3, right?
X times 3, that's 3X,
and if I double that,
that's 6X, all right?
Look a the second one:
X times 5, that's 5X,
and if I double that,
that's 10X.
That happens
with all problems like this,
every single time, okay?
Now, let's look
at that last term.
I see here it's 9.
Look back at
our original problem.
We see the number,
the constant term was 3.
What do we know
about exponents and products?
3 squared is 9, okay;
we know that to be true.
Look at the next one;
5 squared is 25.
So that last term
is always N squared.
Whatever the N was
originally,
it's always that one squared.
And we call this
the sum of squares.
What we're doing right here,
is called a perfect square
trinomial--lot of words.
Basically, you're squaring
a binomial, right?
And your answer,
they call it a perfect square
trinomial,
because it ends up
with three terms.
And it always takes
this format.
So M plus N squared 2
always ends up in this format:
M squared plus 2MN
plus N squared.
Use that shortcut
when you recognize it,
and remember you can always
check your answer with FOIL.
Let's look at when we had
the difference here.
So our answer here,
if we think back,
it was X squared
minus 8X plus 16, right?
Then, our answer for this one
was X squared
minus 14X plus 49.
Because we just did that,
the one, the product,
that perfect square trinomial,
you probably recognize this one
a little bit faster, right?
Because you see these
are in the format
M minus N squared,
but essentially,
the same thing's going on
over here as it was before;
it's just that the sign
is different.
Because our first term here
is the M squared, right?
The second term,
it's still the product
of these two terms.
I'm still multiplying these
together and doubling it.
It's just that
now it's always a minus.
So this is always minus 2MN.
Because X times 4, that's 4X,
double that, that's 8X,
but I was subtracting,
so it's like you're treating
this as a negative 4.
Then, 7--
or X times negative 7--
so that's negative 7X,
and then you're doubling that,
so negative 14X, okay?
The last terms here,
if I look at this,
all right, 4 squared,
that's 16.
7 squared, that's 49.
So it's still the end terms
squared over here, okay?
If you have a difference here,
still a perfect square
trinomial,
you just have a slight
sign change with this one.
Use this shortcut
if you have a problem.
And you can always check
your answer with FOIL, okay?
All right, let's keep going.
Look at this one.
X plus 6 squared
is equivalent to?
We've got to pick
the right answer.
Use the shortcut
and what we know
about that special product.
So, I see
I'm adding here, right?
I know this is in the form
of M plus N squared,
and knowing
what we learned,
I know my answer
should take this format:
M squared plus 2MN
plus N squared, okay?
I look through
my answer choices and see
which one takes that answer
format.
Then I'm good.
So, let's see.
To help me, I'm going to
substitute over here.
I'll say, all right,
that M plus N squared,
that's like X plus 6 squared,
right?
So M squared, that means I need
to square my X,
so X squared.
Now, that middle term should be
2 times M, times N.
Let's do M times N first.
X times 6, that's 6X,
and then I'm doubling that,
so 12X, right?
Then, that N squared
should be 6 squared,
so 6 squared, 36.
Just scan your answer choices
and find that one.
And if you look, it's B.
So you see
how we can use that shortcut
to get to the answer.
Check it with FOIL,
just to be sure.
All right, this one.
Okay, so similar idea,
but this time
we have subtraction.
Okay, so X minus 3 squared
is equivalent to?
So I'm going to write
my M minus N squared,
and then I'll have M squared
minus 2MN plus N squared
because that was the pattern
that we saw before.
Let's figure out
what's going on here.
We've got
our X minus 3 squared;
I'll write that under it
to keep straight.
So, the first thing
I need to is square the M.
In this case, that's X,
so my first term,
that'd be X squared, right?
Next, I need to multiply M
times N together,
double that,
and it's a minus here,
so write that down.
M times N, that would end up
giving me 3X, right?
X times 3 and then I need
to double that,
so if I double 3X,
that's 6X.
Then, N squared,
so that would be 3 squared,
that's 9, okay?
Now, just scan your answer
choices and it's C.
Okay.
We're at the point
where it's your turn to try.
Use what you learned
about special products
and figure out
the right answer.
To check your answer,
press play.

(female describer)
X plus 9, squared,
is equivalent to:
Letter A: X squared
plus 81X plus 18.
Letter B: X squared
plus 18X plus 81.
Letter C:
X squared minus 18X plus 81.
You ready to check?
Let's go.
I see I'm adding here
with my problem--
it's X plus 9, squared.
My answer is going
to take the format...
let's see, M plus N squared--
I'll have M squared
plus 2MN plus N squared.
It helps to write that,
to remember it,
and then it just helps you
keep things organized.
You know what you're following
as you move along.
I have X plus 9, squared;
that's what
I'm working on here.
If I'm following
this pattern,
then I need to square
the first term,
so that's X squared.
I need to multiply
X times 9,
so that's 9X,
and then double that,
so that's 18X.
I need to square the N,
so in this case,
that'd be 9 squared,
which is 81.
Okay, you've got your answer,
now just scan
the answer choices.
It's B, okay?
All right.
To see another way--
because remember, we could
always check with FOIL--
here you go.
Remember when we started,
this means X plus 9,
times X plus 9.
That's all that means.
Okay, so if you FOIL,
then I have X times X,
so that's X squared.
So my outer terms next,
so X times 9,
so that's 9X, right?
My inner terms,
so 9 times X,
that's another 9X, okay?
Then, your last terms,
so 9 times 9, that's 81.
Then, look and see
where your like terms,
like we've been noticing,
they're in the center.
So 9X plus 9X, that's 18X.
So X squared
plus 18X plus 81.
You could always
just throw back to FOIL
to check your answer.
I hope you got
a handle on special products,
and know we could
still use FOIL just to verify
that our answer's correct.
See you back here soon.
Bye!