Welcome to a lesson
on the Distributive Property.
The Distributive Property is often
referred to as multiplication
across addition or subtraction.
The Distributive Property is stated here,
where a times the quantity, b plus c,
equals a times b, plus a times c.
So when we distribute the a,
we have two products, we have a times b,
and then plus, a times c.
Let's begin by verifying this,
for given values of a, b and c.
For example, let's let a equal two,
b equal three and c equal four.
So performing substitution,
we would have two times the quantity, three plus four,
equals, now when we distribute the two,
we'd have two times three, plus two times four.
Let's verify the left side is equal to the right side
by simplifying using the Order of Operations.
On the left side, we simplify inside the parentheses first,
three plus four is equal to seven,
so we'd have two times seven, which is equal to 14.
And on the right side, we would multiply before adding,
two times three equals six,
and two times four equals eight,
and six plus eight is equal to 14.
While this is not a proof,
it does verify the Distributive Property holds true.
There's a couple of things I do want to mention
before we take a look at more examples,
first, if we happen to have subtraction here,
the Distributive Property still holds true.
When we have subtraction,
we have the option of treating the term c
as a negative term, or if we want,
we can just have subtraction between the two products.
And also, if we have a sum or difference
of more than two terms inside the parentheses,
the Distributive Property still holds true,
we just have additional products when distributing.
Now let's look at some additional examples.
We're asked to use the Distributive Property to expand.
Example one we have
five times the quantity two x plus four,
so we'll distribute the five and have two products,
we'll have five times two x, plus five times four.
Again, when we distribute,
we'll have five times two x, plus five times four,
which would be equal to five times two x is 10 x,
plus five times four equals 20.
So here's our product after distributing.
Example two we have negative three
times the quantity x squared, minus two x plus seven.
So because we have three terms inside the parentheses,
when we distribute the negative three,
we'll now have three products.
One,
two,
three.
So our first product is going to be
negative three times x squared,
that'd be negative three x squared.
Now for the next part,
because we have minus two x here,
we'll treat this term as negative two x.
So we have plus negative three times negative two x,
which is positive six x,
so we have plus six x.
And then for the third product
we have plus negative three times positive seven,
which is negative 21.
Instead of writing plus negative 21 though,
we just write minus 21.
So here's our product after distributing.
Now for example three we have negative,
or the opposite of the quantity,
five x to the fourth, minus eight.
So to distribute through the parentheses,
let's re-write this as negative one times
the quantity five x to the fourth minus eight.
And now we'll distribute the negative one.
So our first product is negative one
times five x to the fourth,
that's negative five x to the fourth.
And then we'd have plus negative one times negative eight,
which would be positive eight,
so we have plus eight.
So this would be our product after distributing.
Looking at the given expression again though,
if we think of this negative sign as the opposite,
notice how the opposite of
five x to the fourth, minus eight,
we just change the sign of
the terms inside the parentheses
to get negative five x to the fourth, plus eight.
In example four we have two fifths
times the quantity x divided by four, minus one third.
So we'll distribute the two fifths,
and we'll show some extra work here.
So our first product is going to be
two fifths times x over four, or x divided by four,
and now we find this next product,
I'm going to go ahead and leave this as subtraction,
so we'd have minus two fifths,
times one third.
Notice in this example,
I treated this as minus positive one third,
so we kept the minus sign here,
and we treated this as a positive one third.
Now let's go ahead and find the products.
Before we multiply though, we can simplify here,
there's a common factor of two,
there's one two in two, and two twos in four.
So our first product here is going to be
x divided by 10, and for our second product,
we have two fifteenths.
So while this form is perfectly okay for our product,
we could also write this as
one x over 10, minus two fifteenths.
And then for this first term,
we could write this as one tenth x, minus two fifteenths.
So again,
both this expression and this expression are acceptable,
but we should recognize they are equivalent.
Okay, I hope you found this helpful.