>> Welcome to a video on the distributive property.
The goals are to identify the distributive property and also
to apply the distributive property.
The distributive property states when we multiply a factor and a sum
or difference we multiply the factor by each term of the sum or difference.
So for example if we have a times the quantity b plus c,
so if we have a times the quantity b plus c we're going to multiply the a and the b
and multiply the a and the c. So we'll have a times b plus a times c. Similarly
if we have a times the quantity b minus c we'll multiply the a and the b and also multiply the a
and the c, so we'll have a times b minus a times c. Some times you'll see the distributive
property written using this form here in red and it's just a way to combine these two rules
into one, notice we have a plus or minus sign meaning if we used a plus sign here
on the left we'll use the plus sign here on the right and if we have a minus sign
on the left we'll use a minus sign on the right.
Let's go and take a look at some examples,
we want to rewrite each expression using the distributive property in order to simplify.
So in this first example we have 5 times the quantity 6 plus 2 so we'll multiply the 5
and the 6 and also multiply the 5 and the 2.
So we'll have 5 times 6 plus 5 times 2.
We have now applied the distributive property and now we will continue to simplify.
Well, 5 times 6 would be 30 and 5 times 2 would be 10, 30 plus 10 is equal to 40.
We have simplified this expression using the distributive property.
Well, let's take a look at this again, if we had 5 times the quantity 6 plus 2 and we wanted
to simplify using the order of operations we would perform the addition inside the
parentheses first, so this would 5 times 6 plus 2 that would be 8 and 5 times 8 is also 40.
So whether we applied the distributive property to simplify or use the order of operations
of course the result is the same.
Let's take a look at another one where we have a variable involved,
here we have 2 times the quantity x minus 4, so we're going to distribute the 2
or multiply the 2 and the x and also multiply the 2 and the 4.
So we'll have 2 times x minus 2 times 4, well 2 times x is 2 x
and 2 times 4 is 8 so we have 2 x minus 8.
Let's go ahead and take a look
at the distributive property geometrically meaning let's say that the length
of this side is 4 inches and the length of this entire side here is 15 inches.
If we wanted to break this area into two smaller rectangles with this black line here let's say
that this length here is 5 inches and this length over here would be 10 inches.
We can use this to model the distributive property,
we know the area of a rectangle is length times width,
so this larger rectangle has the dimensions of 4 inches by 15 inches
which means the area would be 4 times 15 or 60 square inches.
Now just to illustrate the distributive property we could take this product here
which represents this area in green and write it as 4 times 5 plus 10 instead of using 15,
again if this total length here is 15 and if we broke this length up into two pieces,
one piece being 5 and one piece being 10 this result here would be the same.
Now if we apply the distributive property we'd have 4 times 5 and 4 times ten,
well 4 times 5 is 20, 4 times 10 is 40 and the result would be 60 square inches.
Well, this 20 here represents the area of this smaller rectangle here, notice the dimensions
of this rectangle are 4 inches by 5 inches and this 40 here represents the area
of this rectangle here with dimensions 4 inches by 10 inches,
so the area would be 40 inches squared.
So this illustration just justifies the distributive property geometrically.
Let's go and take a look at two more examples,
here we want to rewrite these expressions using the distributive property.
So on this first example we'll distribute the 9 meaning we'll multiply 9
and 2 x and also multiply 9 and 7.
So we'll have 9 times 2 x plus 9 times 7 and we'll go ahead and simplify this,
9 times 2 x would be 18 x and 9 times 7 would be 63.
On the last example we'll distribute the 3 x so we will have the product of 3 x
and x and the product of 3 x and 4.
So we'll have 3 x times x minus 3 x times 4.
Well, 3 x times x would be 3 x squared minus 3 x times 4 we can multiply the 3 and the 4
so we'd have 12 x. that will do it for this video, thank you for watching.
[ Silence ] ------------------------------a75ec279a249--