Hey, guys,
welcome to Algebra I.
Today's lesson
focuses on finding
the product of a binomial
with a trinomial.
The distributive property helps
you get through these problems.
You ready to get going?
Let's go.
Okay, to help you understand
how to multiply
a binomial times a trinomial,
I first need to throw back
to multiplying two binomials.

(female describer)
On the screen:
Find the product.
x plus 3 times x plus 7.

(teacher)
We've been using FOIL
as a method to get through
these problems.
FOIL was just a shortcut around
the distributive property.
It is the distributive
property in action
without drawing all
of those loops and hoops.
Let's think back.
Using FOIL, you multiply
the first two terms together.
Right?
So, x times x,
which gives us x squared.
Then we multiply
the outer terms together,
so that was x times 7--
in this case, 7x.
Okay?
We'd go to the 3,
in this case--
multiply 3 times x,
those inner terms: 3x.
Then we multiply the 3
times the 7, the last terms,
and get 21.
So FOIL is the same thing
as the distributive property.
That x squared,
that came from multiplying
x times x.
That was where
the "F" came from.
Then we multiplied
or distributed x to the 7,
multiplying those together.
Those were our outer terms.
The 3x came from multiplying
these together,
or distributing
the 3 to the x.
That's where
that term came from.
Then the last term came from
distributing the 3 to the 7.
Okay?
Essentially,
FOIL is the same thing
as the distributive
property.
To finish
this problem off,
we have some like terms
we can combine--
7x plus 3x gives us 10x.
So, x squared
plus 10x plus 21.
We'd use FOIL
to get that answer.
The problem arises
when you're not multiplying
two binomials together--
like here, you're multiplying
a binomial times a trinomial.

(female describer)
x plus 3 times 2 x squared
plus 4x minus 1.
In that case,
you can't use FOIL
because you don't
have two binomials,
so you use
the distributive property.
There's not really a shortcut
around this one, unfortunately.
You have to distribute
to get to your answer, okay?
I start by drawing the hoops
to keep us straight.
If you don't need them after
a while, stop writing them.
I'm going to take the x,
and I'm going
to distribute it to each
term in my trinomial.
I'm ignoring
this 3 for a second.
I pretend like the 3
is just not even there.
So, x times 2x squared--
I'll show some work down here.
Okay, that's 2x
to the 3rd power.
Now I have x times 4x.
So x times 4x.
That's 4x squared, right?
So, plus 4x squared.
Then x times negative 1.
So x times negative 1.
So, that's negative 1x, right?
So minus x.
You've handled distributing
the x throughout these terms.
Now you have
to distribute your 3
to each of these terms
as well.
That's how we get
through this problem.
I will get rid of this work
so we have
a little room to work, okay.
Okay.
Now I handle the 3.
I'm going to distribute that 3
to each term in my trinomial.
So I've got
3 times 2x squared.
That's 6x squared, okay?
I'm going to write that
right up here, 6x squared.
Now I have 3--
Oop, get that pen back;
it escaped us somehow.
All right, there we go.
Okay, so 3 times 4x, right?
3 times 4x.
Well, that's 12x, right?
So I've got
plus 12x right up here.
I need to distribute that 3
to this term, that negative 1.
So, 3 times negative 1.
That's negative 3, right?
So where I'm writing my
answers--lost that a second--
I'd have minus 3.

(female describer)
She's rewritten the problem
as 2x cubed plus 4x squared
minus x plus 6x squared
plus 12x minus 3.
Now you're done distributing
everything in your problem.
You're at the point
where you combine like terms.
I get rid
of this scratch work
so we can just focus
on combining like terms.
I'll get rid
of those hoops as well.
Okay.
Let's see where we can
combine like terms here.
We're writing the answer
in standard form.
We start with the exponent
with the largest degree
and work our way down.
We'll start with
the largest degree,
the biggest exponent,
basically.
So my largest exponent
in this problem is 3,
and I have one term that's
raised to the 3rd power.
There's nothing
to combine that with.
That will just stay
2x to the 3rd.
I scratch it out to hint
I've dealt with it.
So now I'll look
for exponents of 2,
and you scan your terms
and see you have two.
You have 4x squared,
and you have 6x squared.
So 4x squared
plus 6x squared--
that's 10x squared, right?
Okay, so plus 10x squared.
I've handled this now.
So, now let's look
for powers of 1.
I see I have a negative x
and a positive 12x.
So combine those.
So, negative x plus 12x,
that's 11x, okay?
I've dealt with that.
I have this constant term
on the end, this minus 3.
It's my only constant term.
There's nothing
I can combine it with.
So I'll just write
minus 3.
And after all that work,
you're all done.

(female describer)
2x cubed plus 10x squared
plus 11x minus 3.
The key to that is using
the distributive property,
keeping everything
really organized,
and working your way
through the problem.
Let's try another one.
All right,
let's bring this up a bit.
Okay, so I have x minus 5
times x squared
minus 3x plus 2.
You remember your best friend,
the distributive property.
That's how we get
through this.
I ignore this minus 5
for a minute--
pretend like it's not there--
and just focus my attention
on distributing this x.
So I'm going to multiply
x times x squared.
I'll do some work here.
So x times x squared--
that's x cubed, right?
So x cubed.
Then I'll have x
times negative 3x,
so x times negative 3x.
That's negative 3x squared,
right?
So minus 3x squared.
I distribute this x
to this positive 2.
Can't leave it out.
So x times 2--
that's 2x, okay?
So plus 2x.
And you are all done
distributing that first term
in that binomial, that x.
So now you distribute
that negative 5
to each of these terms
in this trinomial.
I'm going to give us room
for scratch work.
I show you my work
so you know what's going on,
how I'm getting
these answers.
If you can handle
that mental math
and just write
your answers out, feel free.
Let's distribute
that negative 5
throughout those terms.
So, negative 5
times x squared...
That's negative 5x squared,
right?
So up here,
I'll have minus 5x squared.
I distribute negative 5
to the next term,
to this negative 3x.
So negative 5
times negative 3x.
Negative 5 times negative 3x.
So, that is 15x, right?
So, plus 15x.
And then the last term.
Now I need to distribute
this negative 5
to this positive 2.
So negative 5 times 2--
that's negative 10.
So, last term in my row--
I keep moving our screen
out of the way--minus 10.
Okay, we're at the point
where we combine
like terms
and clean this problem up.
Let's get some room
to work here.
Get scratch work
out of the way.
You're working on paper;
don't erase it.
But I want you to have
a clear view.
That's why
I erase my work.
Okay, remember, I'm starting
with the largest exponent.
I want my answer
written in standard form.
So, I scan my terms--
the largest exponent's a 3.
There are no terms
besides this first one raised
to the 3rd power,
so I just carry that along.
It's x cubed.
I'll cross it out
so I know I've handled it.
Working my way down
as far as exponents,
I look for exponents with
a value of 2, and I see two.
I have negative 3x squared
and negative 5x squared.
So negative 3x squared
minus 5x squared--
that's negative 8x squared.
So I'm cool
with that one now.
I look for exponents of 1--
remember, you don't
see exponents of 1--
so I know where
they have to be hiding.
I have 2x and 15x--
so 2x plus 15x,
that's 17x, okay?
I'll cross them out
because I've handled them.
The last remaining term
is this constant term,
this negative 10,
so I know I'll have minus 10
because there's nothing else
I can combine it with.
You look
at your masterpiece.
You're all done.
All right?
Okay, let's try
one more together.

(female describer)
x plus 1 times x squared
minus 4x plus 3.
All righty, remember,
we're using
the distributive property
to get through these.
So I'm going to ignore
that 1 to start with
and just focus
on x at first.
I'm going to distribute this x
throughout each of these terms.
So, x times x squared--
show my work down here.
That's x cubed, okay?
I have x to the 3rd power.
Now I need to distribute this x
to this negative 4x.
So x times negative 4x--
That's negative 4x
squared, okay?
So, minus 4x squared.
Now I need to distribute
this x to this 3.
So, x times 3.
x times 3,
that's 3x.
So, plus 3x, okay?
I'm completely done
distributing the x.
I've worked it
through each term.
I need to multiply
each term by 1
so that I can
distribute that 1.
Let's get this work
out of the way
so we have some more room.
Get the pen back,
all right.
We have 1 times x squared;
we need to handle that first.
So, 1 times x squared--
that's x squared.
Okay?
Now we have 1
times negative 4x.
So, 1 times negative 4x--
that's negative 4x.
So, minus 4x.
Then we have 1 times 3
to get it to that last term--
1 times 3; that's just 3.
Not surprising
that no terms change
because multiplying by 1
doesn't change anything, okay?
Now we're done
distributing.
We need to combine
like terms, right?
Remember, this is
where I get us
some more room to work.
All righty,
get the pen back, okay.
I look for my largest
exponent first.
I want my answer
in standard form.
The largest exponent is 3.
There's no other terms
that have an exponent of 3,
so I'll carry that along.
So x cubed.
I've handled it;
I'll cross it out.
I look for exponents of 2.
I have this negative 4x squared
and this positive x squared.
So, negative 4x squared
plus x squared--
that's negative 3x squared,
so minus 3x squared.
You've handled those;
let's cross them out.
All righty, so now
we look for x--
terms that just have x.
I see I have 3x;
I have this negative 4x.
So, 3x minus 4x,
that's negative 1x.
So I'll write negative x.
I'll cross those out
because I'm all done.
I have a constant term,
the only term left.
There's nothing
to combine it with.
So plus 3,
and you're all done.
You've reached
your answer.
It's already
in standard form for you.
Okay?
All right, now it's
your turn to try.
Press pause,
take a few minutes.
Work through
these problems.
When you're ready to compare
your answers, press play.

(female describer)
Find each product.
1. x minus 4 times 2x squared
plus x minus 1.
2. x plus 6 times x squared
minus 3x plus 2.
Let's see what you got.
For the first one,
x minus 4 times 2x squared
plus x minus 1--
You should have 2x cubed
minus 7x squared
minus 5x plus 4.
A lot of terms,
but that should be your answer.
The number 2, x plus 6
times x squared
minus 3x plus 2--
Your answer there
should have been
x cubed plus 3x squared
minus 16x plus 12, okay?
If you need to see
how I got those,
stay with me,
and I'll show you.
So just like
we've been doing,
it's all about
the distributive property.
Start with the first term
in your binomial
and pretend like second term
is just not there.
Multiply that first term times
each term in that second group.
So x times 2x squared...
That's 2x cubed.
All right, so I know
the first term, 2x cubed.
Now I have x times x, right?
So x times x;
that's just x squared.
So, plus x squared.
Then the last term
in this one,
x times negative 1.
That's just negative 1x.
I'll just write negative x,
so I'll bring that up here.
You're all done
distributing that first term.
You've got to do the same thing
with the second one.
I'm going to get us
some space up here.
All right,
let's handle that second one.
Multiply negative 4 times
each term in the second group.
So, negative 4
times 2x squared...
That is negative 8x squared.
I'll just bring that up here,
so minus 8x squared.
Now I have
negative 4 times x,
distributing it
to the second term.
So negative 4 times x--
that's negative 4x, right?
So, just minus 4x.
Then the last term,
negative 4 times negative 1--
negative 4 times negative 1.
That's a positive 4.
So, my last term
over here is plus 4.
We've got
to clean it up.
We've got to combine like terms;
get it all nice.
Let's get this--
get the--
Get that out of our way,
even those out,
so we can just focus
on combining like terms.
Look for the largest
exponent first
because you want your answer
in standard form.
The largest exponent is 3--
the only term
that has an exponent of 3--
so 2x cubed, okay?
Cross it out
because I've handled it.
Look for exponents of 2.
I see I have x squared
and negative 8x squared.
If I combine those,
I get negative 7x squared,
so minus 7x squared.
Cross them out
because I've handled them.
Next I see I have
negative x and a negative 4x.
When I combine those,
I get negative 5x, okay?
Cross them out
because I've handled them.
The last term left
is 4--
nothing left to combine
it with, so plus 4.
All right, and that's how
I got that first answer.
It's all about
the distributive property
and then just combining your
like terms when you're done.
Let me show how
I got the second one.
I did the same thing,
distributive property.
So x times x squared, right?
So x times x squared--
that's x cubed, okay?
So, I know that
was my first term.
Now I have x
times negative 3x,
so, x times negative 3x.
That's negative 3x squared,
right?
So, minus 3x squared.
Then x times 2
to get to that last term.
So x times 2;
that's just 2x.
So, plus 2x.
You've handled that first term,
distributing it throughout.
So now let's handle
the second one.
Let's get us
some space up here, okay.
I'm going to handle the 6.
So, 6 times x squared.
That's 6x squared, right?
So plus 6x squared.
Now I need to multiply
6 times that negative 3x
to keep working
through over there.
So 6 times negative 3x;
that's negative 18x.
So, minus 18x--
squeezing it in over there.
And now I need to
multiply the 6 times the 2.
So 6 times 2, that's 12.
Oh, can I make it fit?
So plus--
I cannot--plus 12.
We can combine
like terms.
We just need
to clean this up.
All right, let's get this
out of our way.
Okay, let's clean
this up.
You start looking for your
largest exponent first, right?
I have one term
with an exponent of 3,
the largest one--
can't combine it
with anything.
So, just x cubed.
And I'll cross it out
to note I've handled it.
I look for exponents of 2.
Negative 3x squared
plus 6x squared--
that's a positive 3x squared.
All right, done;
cross those out.
Now 2x and negative 18x,
combining those,
I'd get negative 16x,
so minus 16x.
Cross it out
because I've handled it.
Your last term is that
constant number, that 12.
Nothing left to
combine it with, so plus 12.
The magic has happened.
You're all done, okay?
I hope you're
feeling good knowing
how to multiply a binomial
times a trinomial,
and you saw how important
the distributive property is
to those problems.
See you soon.
Bye, guys.