Hi, I’m Rob. Welcome to Math Antics!
In this lesson, we’re going to talk about the Distributive Property, which is a really useful tool in Algebra.
And if you watched our video called The Distributive Property in Arithmetic,
then you already know the basics of how the Distributive Property works.
The key idea is that the Distributive Property allows you to take a factor
and distribute it to each member of a group of things that are being added or subtracted.
Instead of multiplying the factor by the entire group as a whole,
you can distribute it to be multiplied by each member of the group individually.
And in that previous video, we saw how you can take a problem like:
3 times the group (4 + 6) and simplify it two different ways.
You could either simplify what was in the group first,
OR you could use the distributive property to distribute a copy of the factor ‘3’ to each member of the group,
and no matter which way you go, you get the same answer.
But in Algebra, things are a little more complicated, because we aren’t just working with known numbers.
Algebra involves unknown values and variables, right?
So in Algebra, you might have an expression like this: 3 times the group (x + 6).
In this expression, we don’t know what value ‘x’ is.
It could be ‘4’ like in the last expression, but it doesn’t have to be.
It could be ANY number at all! And since we don’t know what it is, that means we CAN’T simplify the group first in this case.
Our only option here is to either leave the expression just like it is and not simplify it at all,
OR to use the Distributive Property to eliminate the group.
Just like in the arithmetic video, we can distribute a copy of the ‘3 times’ to each member of the group
so the group goes away and we end up with 3 times ‘x’ plus 3 times 6.
The 3 time ‘x’ can’t be simplified any further because we still don’t know what ‘x’ is,
but we can simplify 3 times 6 and just write 18.
So the distributed form of this expression is: 3x + 18
And even though we can’t simplify these expressions all the way down to a single numeric answer without knowing the value of ‘x’,
we do know that these two forms of the expression are equivalent because they follow the distributive property.
So the Distributive Property works exactly the same way whether your working with numbers or variables.
In fact, in Algebra, you’ll often see the Distributive Property shown like this:
‘a’ times the group (b + c) equals ab + ac
Or you might see it with different letters, like x, y, and z, but the pattern will be the same.
This pattern is just telling you that these two forms are equivalent.
In the first form, the factor ‘a’ is being multiplied by the entire group.
But in the second form, the factor ‘a’ has been distributed so it’s being multiplied by each member of the group individually.
And if you’re looking at this thinking, “what multiplication?”,
remember that multiplication is the ‘default’ operation which is why we don’t have to show it in this pattern.
Since the ‘a’ is right next to the group, it means it’s being multiplied by the group,
and on this other side, since the copies of the ‘a’ are right next to the ‘b’ and ‘c’, it means they are being multiplied also.
And even though this pattern is usually shown with addition in the group,
remember that it also works for subtraction since subtraction is the same as ‘negative’ addition.
But the distributive property does NOT apply to group members that are being multiplied or divided.
Okay, so this is the basic pattern of the Distributive Property.
It’s usually just shown with two members in the group, but remember that it works for groups of any size.
We could have ‘a’ times the group (b + c + d) and the equivalent ‘distributed’ form would be: ab + ac + ad
Here’s a few quick examples that have a combination of numbers and variables to help you see the patterns of the Distributive property:
2 times the group (x + y + z) can be changed into the distributed form: 2x + 2y + 2z
10 times the group (a - b + 4) can be changed into the distributed form: 10a - 10b + 10 times 4 (which is 40).
And… ‘a’ times the group (x - y + 2) can be changed into the distributed form: ax - ay + a2 (or 2a which is more proper).
So whether you‘re dealing with numbers or variables or both,
the key concept is that the factor outside the group gets distributed to each term in the group.
Each TERM in the group?
But… I thought “terms” were parts of polynomials,
and I thought we were WAY past all that by now!
Ah - I was hoping you would notice that.
And in fact, the members of these groups really are just simple terms in a Polynomial.
Well… that’s what I’m here for… noticing things.
Ooooo! - A butterfly!!
Realizing that these groups of things being added or subtracted are really just Polynomials
will help you see why the Distributive Property is SO useful in Algebra.
For example, in this simple expression: 2 times the group (x + y)…
the ‘x’ and the ‘y’ are simple terms in the polynomial x + y.
Each of the terms has a variable part but no number part.
And if we apply the Distributive Property to the group, we get the equivalent form: 2x + 2y
But what if the polynomial was just a little bit more complicated? …like this: 2 times the group (3x + 5y)
In this expression, each of the terms in the polynomial DOES have a number part that is being multiplied by the variable part.
But we can still use the Distributive Property to distribute a copy of the factor ‘2’ to each term in the polynomial.
Wait just a second here!
I NOTICED earlier that you said the Distributive Property does NOT work with members of a group that are being multiplied,
and I also NOTICED that these terms DO have multiplication.
What’s up with that?
Ah… That’s a good question!
And it can be a little confusing to see how it all works at first.
But notice that even though the terms do have multiplication in them, the terms THEMSELVES are being added.
So we distributed a copy of the factor ‘2’ to each whole term, but NOT to each part of a term.
In other words, we treat each term in a polynomial as a individual member of the group,
even if that term has multiplication going on inside of it
(which is common since there is often a variable part and number part being multiplied together).
Getting back to our example… Distributing the factor ‘2’ to each term gives us ‘2’ times ‘3x’ plus ‘2’ times ‘5y’.
But this can be simplified even further because we know that 2 times 3 is just 6 and 2 times 5 is just 10.
So the distributed form is: 6x + 10y
Let’s try another example of a factor times a polynomial: 4 times the group (‘x squared’ + 3x - 5).
First we need to identify the terms of this polynomial
so when we distribute the factor, we just make one copy of it for each term.
This polynomial has three terms: ‘x squared’, positive 3x and negative 5.
So we distribute a copy of the factor ‘4’ to each term and we get:
4 times ‘x squared’ (or just 4 ‘x squared’)
4 times 3x which is 12x (since 4 times 3 is 12),
and 4 times the negative 5 which is negative 20.
So the equivalent distributed form is: 4 ‘x squared’ + 12x - 20
Let’s see another example: ‘x’ times the group (‘x squared’ - 8x + 2)
In this expression, the factor being multiplied by the group is actually a variable,
but the Distributive Property works exactly the same way.
And it says we can distribute that factor and multiply it by each term of the group individually.
The first term is ‘x squared’ (which is the same as ‘x’ times ‘x’)
so if we multiply that by ‘x’, we’ll get ‘x-cubed’ since that would be three ‘x’s multiplied together.
The next term is negative 8x so if we multiply that by ‘x’ we’ll have negative 8 times ‘x’ times ‘x’
which is the same as negative 8 ‘x squared’.
Last of all we have the term positive 2, and ‘x’ times positive 2 is just 2x,
so after distributing the factor ‘x’ to each member of the original group, we have the polynomial:
‘x cubed’ - 8 ‘x squared’ + 2x
See why the Distributive Property is so handy in Algebra?
It shows us how to multiply a polynomial by a factor!
We just distribute a copy of that factor to each of the polynomial’s terms.
So I know what you’re thinking…
if we can distribute something to each member of a group…
Can we do the process in REVERSE and UN-distribute something?
We sure can!…
Take a look at this polynomial: 4 ‘x cubed’ + 4 ‘x squared’ + 4x
Notice that each term of this polynomial has a factor of ‘4’ as its number part.
In fact it kinda looks like someone distributed a factor of 4 to each term.
Since distributing a factor means making multiple copies of it for each member of a group,
UN-distributing is going to mean
consolidating multiple copies of a factor
into a single copy that is multiplied by the whole group.
So in this case, we can remove the factor of '4' that is being multiplied by each term individually,
and then we can consolidate those into a single factor of '4' that is being multiplied by the entire polynomial
by using parentheses to turn the polynomial into a group.
But mathematicians usually don’t call this “UN-distributing a 4”.
Instead they would say that we “factored out a 4” from the polynomial.
So you can use the Distributive Property both ways.
If you get the expression ‘a’ times the group (b + c), you can distribute a copy of the factor ‘a’ to each member of the group.
But if you’re given the expression, ab + ac, you can apply the Distributive Property in reverse
and “factor out” the ‘a’ so that it is multiplied by the whole group at once.
It’s important to realize that neither of these changes the value of the expression.
Distributing and UN-distributing a factor are just ways of going back and forth between two equivalent forms of an expression.
And it works in cases where it’s not quite so obvious too.
For example, Look at this polynomial: 8x + 6y + 4z.
Notice that each of the number parts of this polynomial is an ‘even’ number which means it contains a factor of ‘2’.
8 is 2 times 4
6 is 2 times 3
and 4 is 2 times 2
So each of these terms has a common factor of ‘2’ and that means that if we want to, we can factor out that ‘2’.
We can apply the Distributive Property in reverse!
We remove the ‘2’ from each term and consolidate it to form a single factor that’s multiplied by the whole polynomial at once.
And it works exactly the same way for variables too.
What if we have the polynomial: ‘a’ ‘x squared’ + ax + a
Each of these terms has the common factor ‘a’ so you could UN-distribute or “factor out” the ‘a’.
Notice that when we do that to the last term (which was just ‘a’) that term becomes a ‘1’
because there is always a factor of ‘1’ being multiplied by any term.
Alright, so that’s the basics of how the Distributive Property works in Algebra.
As you can see, it can get pretty complicated for big Polynomials,
but the most important thing is to understand how it works in simple cases so you can build on that understanding in the future.
Being able to recognize the pattern of the Distributive Property and to apply it both directions
will allow you to rearrange algebraic expressions and equations when you need to.
And remember… the key to really understanding math is to try working some practice problems
so that you actually use what you’ve learned in the video.
As always, thanks for watching Math Antics and I’ll see ya next time.
Learn more at www.mathantics.com