The Distributive Property

Uploaded by MuchoMath on 01.03.2009

>> Professor Perez: Hey!
This is Professor Perez again.
Today, we're going to look at the distributive property.
Now we've looked at it before when we introduced it, but now we're going to use it
in some problems that are a bit more involved.
And of course, we've got to get Charlie out.
He better be ready to go!
Charlie, what are you doing over there?
Quit fooling around!
We're doing the distributive property.
>> Charlie: What?
>> Professor Perez: Yeah, that's right!
Now, remember, you don't learn this now, you can always come back and do it...
>> Charlie: What?
>> Professor Perez: Next semester!
Uh-huh! All right, here we go, right there, Charlie.
Now, we have negative 3 times the quantity x plus 4.
Order of Operations says we're supposed to do the parenthesis first,
but you cannot add x plus 4...unless you'd like to repeat the class...with him.
>> Charlie: What?
>> Professor Perez: So, what we have to do,
is we have to use the distributive property, Charlie.
We're going to distribute that negative 3 by multiplication into the parenthesis.
And so, we have negative 3 times x, which is negative 3x,
and plus a negative 3 times 4, which is a negative 12.
And remember, adding a negative number is the same as subtracting,
so our answer is negative 3x subtract 12.
And I know there's some of you that can skip that middle step,
but as long as you can explain why, you're fine, and I'm just showing you why.
Okay, let's do another problem.
So, here's one of those problems where people like to use a shortcut, but before we talk
about the shortcut, let me go through the steps.
Okay, again, we're supposed to do parenthesis first, but we can't do it,
so we're going to distribute that 3 through.
Now, a lot of people like to view that subtract 3 as a negative 3.
You can, because remember, adding a negative 3 is the same as subtracting 3.
And so, if we bring down the 2, we can distribute that negative 3 through,
and say negative 3 times x is negative 3x and negative times 4 is a negative 12, right?
And remember, adding a negative number is the same as subtracting, or you can visualize it
as distributing a 1 through the parenthesis.
And you get 2 subtract 3x subtract 12.
Now, I know, some of those...some
of you say those two middle steps maybe throw you off a little bit
because you are used to just using the shortcut.
So, if we take that out, that's fine.
Some of you can go straight from that step to there, and that's really where we're working to.
You can do that as long as you can explain it somebody as to why you can do that.
So, once we get through that then, we look at our like terms, 2 subtract 12, right?
Which is negative 10 subtract 3x, or we generally put the variable first,
and we have negative 3x subtract 10.
Any answer...either of those two answers is correct.
Now, let's do it the fast way, or the Kung-Fu way.
Some of you say, oh, 2 subtract 3 times x plus 4.
Well, it's 2, and then immediately you go, negative 3 times x is a negative 3x, yes,
because you're subtracting 3x, and negative 3 times a positive 4 is negative 12
and there it is.
You end up with the exact same answer as we had before.
So, as long as you can explain it, you can do it.
Remember, you have to know what you're doing before you can start Kung-Fu-ing.
Anyway, here we go, Charlie.
3 plus 4, times 3x plus 1, Charlie.
Okay, now, we're going to distribute that plus 4, through the parenthesis, 4 times 3x is?
>> Charlie: 12x.
>> Professor Perez: And 4 times 1?
>> Charlie: 4.
>> Professor Perez: There you go.
Now combine our like terms, generally we'll bring our variable first, and what's 3 plus 4?
>> Charlie: 7.
>> Professor Perez: 7.
And there you go.
12x plus 7.
Here we go, now, we have 2 times the quantity x subtract 1 plus 3 times x plus 1.
Remember, we can't do the parenthesis first,
but we have to do the multiplications before we do the addition,
and so our multiplication requires the distributive property.
So we go, 2 times x, and then we'll do 2 times a negative 1.
And then we'll do a positive times x and a positive 3 times 1.
And so, here we go, 2x subtract 2, plus 3x plus 3.
You getting this, Charlie?
Yeah, you better get it, or you're gonna get it!
Anyway, here we go, Charlie, let's combine our like terms, we have the 2x and the 3x,
and the negative 2 and the plus 3, and so what's 2x plus 3x, Charlie?
>> Charlie: 5x.
>> Professor Perez: Negative 2 plus 3?
>> Charlie: 1.
>> Professor Perez: Very nice and go ahead and box your answer.
There you go.
Let's do another one.
Here we go, Charlie, now don't get scared.
We're going to do the same procedure that we've been using.
We're going to take the 2 times the 2x and the 2 times the negative 1,
and the negative 4 times a 3x and the negative 4 times a negative 4.
Okay, now, Charlie, 2 times 2x...
>> Charlie: Is 4x.
>> Professor Perez: Is 4x.
Now, 2 times a negative 1?
Is negative 2.
Negative 4 times 3x?
>> Charlie: 12x...
>> Professor Perez: Negative 12x.
>> Charlie: Negative.
>> Professor Perez: And negative 4 times negative 1 is...
>> Charlie: Plus 4.
>> Professor Perez: Is plus 4.
It works. Remember, those negative numbers, we're visualizing them
as being adding negative numbers.
So you kind of got to think about that.
That's why this works.
And now, remember, Order of Operations says to go left to right, but we can get around that
by visualizing everything as being added and so our like terms are our 4x and a negative 12x,
and a negative 2 and a positive 4.
And now, Charlie, what's 4x plus a negative 12x?
>> Charlie: Negative 8x.
>> Professor Perez: And a negative 2 plus 4?
>> Charlie: Plus 2.
>> Professor Perez: Very nice there Charlie!
Our answer is negative 8x plus 2.
And that is our lecture on the distributive property, so, keep up, keep working,
and keep doing your homework and we'll see you again soon!