Algebra Basics: The Distributive Property - Math Antics - Free Educational videos for Students in K-12

Algebra Basics: The Distributive Property - Math Antics - By Mathantics

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00:03	Uh huh . Hi , I'm rob . Welcome to
00:07	Math Antics . In this lesson , we're going to
00:09	talk about the distributive property , which is a really
00:12	useful tool in algebra . And if you watched our
00:14	video called the distributive property and arithmetic , then you
00:17	already know the basics of how the distributive property works
00:20	. The key idea is that the distributive property allows
00:23	you to take a factor and distribute it to each
00:26	member of a group of things that are being added
00:28	or subtracted , instead of multiplying the factor by the
00:31	entire group as a whole , you can distribute it
00:34	to be multiplied by each member of the group individually
00:37	. And in that previous video we saw how you
00:40	can take a problem like three times the group ,
00:42	four plus six and simplify it . Two different ways
00:45	. You can either simplify what was in the group
00:47	first , or you could use the distributive property to
00:51	distribute a copy of the factor three to each member
00:54	of the group , no matter which way you go
00:56	, you get the same answer . But in algebra
00:59	things are a little more complicated because we aren't just
01:01	working with known numbers , algebra involves unknown values and
01:05	variables . Right ? So in algebra you might have
01:07	an expression like this three times the group X plus
01:11	six in this expression we don't know what value X
01:14	. Is . It could be for like in the
01:16	last expression , but it doesn't have to be ,
01:19	it could be any number at all . And since
01:21	we don't know what it is , that means we
01:23	can't simplify the group first . In this case ,
01:25	our only option here is to either leave the expression
01:28	just like it is and not simplify it at all
01:31	, or to use the distributive property to eliminate the
01:34	group , just like in the arithmetic video , we
01:37	can distribute a copy of the three times to each
01:40	member of the group . So the group goes away
01:42	and we end up with three times X plus three
01:45	times six . The three times X can't be simplified
01:48	any further because we still don't know what exes ,
01:51	but we can simplify three times six and just write
01:54	18 . So the distributed form of this expression is
01:57	three X plus 18 . And even though we can't
02:00	simplify these expressions all the way down to a single
02:03	numeric answer without knowing the value of X , we
02:06	do know that these two forms of the expression are
02:09	equivalent because they follow the distributive property . So the
02:13	distributive property works exactly the same way whether you're working
02:16	with numbers or variables . In fact , in algebra
02:19	, you'll often see the distributive property shown like this
02:22	eight times the group B plus C equals A ,
02:25	B plus A . C . Or you might see
02:28	it with different letters like X , Y and Z
02:30	. But the pattern will be the same . This
02:32	pattern is just telling you that these two forms are
02:35	equivalent . In the first form , the factor A
02:37	. Is being multiplied by the entire group . But
02:40	in the second form , the factory has been distributed
02:43	, so it's being multiplied by each member of the
02:45	group individually . And if you're looking at this ,
02:47	thinking what multiplication , remember that multiplication is the default
02:51	operation , Which is why we don't have to show
02:53	it in this pattern . Since the A . Is
02:55	right next to the group , it means that it's
02:57	being multiplied by the group . And on the other
02:59	side , since the copies of the A . R
03:01	. Right next to the B and C , it
03:03	means they're being multiplied also . And even though this
03:06	pattern is usually shown with addition in the group ,
03:09	remember that ? It also works for subtraction , since
03:11	subtraction is the same as negative edition , but the
03:14	distributive property does not apply to group members that are
03:18	being multiplied or divided . Okay , so this is
03:21	the basic pattern of the distributive property . It's usually
03:24	just shown with two members in the group . But
03:26	remember that it works for groups of any size .
03:29	We could have A times the group B plus C
03:32	plus D , and the equivalent distributed Form would be
03:35	A B plus A C plus A D . Here's
03:39	a few quick examples that have a combination of numbers
03:41	and variables . To help you see the patterns of
03:43	the distributive property . Two times the group X plus
03:47	Y plus C . Can be changed into the distributed
03:50	form two X plus two , Y plus two Z
03:54	. 10 times the group A minus B plus four
03:57	can be changed into the distributed form 10 a minus
04:01	10 B plus 10 times four , which is 40
04:05	and eight times the group X minus Y plus two
04:09	can be changed into the distributed form a x minus
04:13	A . Y plus a two or two . A
04:16	. Which is more proper . So whether you're dealing
04:20	with numbers or variables or both . The key concept
04:23	is that the factor outside the group gets distributed to
04:26	each term in the group , each term in the
04:28	group . But I thought terms were parts of polynomial
04:32	. I thought we were way past all that by
04:34	now . I was hoping you would notice that .
04:36	And in fact the members of these groups really are
04:39	just simple terms and a polynomial . Well , that's
04:42	what I'm here for . No , the same things
04:45	. Oh , how butterfly realizing that these groups of
04:50	things being added or subtracted are really just polynomial .
04:54	Will help you see why the distributive property is so
04:57	useful in algebra . For example , in this simple
05:00	expression , two times the group X plus Y .
05:03	The X and the Y . Are simple terms in
05:06	the polynomial , X plus Y . Each of the
05:08	terms has a variable part but no number part .
05:11	And if we apply the distributive property to the group
05:14	, we get the equivalent form two X plus two
05:16	Y . But what if the polynomial was just a
05:20	little bit more complicated like this ? Two times the
05:23	group three X plus five Y . In this expression
05:26	, each of the terms in the polynomial does have
05:28	a number part that is being multiplied by the variable
05:31	part . But we can still use the distributive property
05:34	to distribute a copy of the factor to to each
05:36	term in the polynomial . Wait just a second here
05:40	. I noticed earlier that you said that the distributive
05:43	property does not work with members of a group that
05:46	are being multiplied . And I also noticed that these
05:49	terms do have multiplication . What's up with that ?
05:53	Uh That's a good question . And it can be
05:55	a little confusing to see how it all works at
05:57	first , but notice that even though the terms do
06:01	have multiplication in them , the terms themselves are being
06:04	added . So we distribute a copy of the factor
06:06	to to each whole term , but not to each
06:09	part of a term . In other words , we
06:12	treat each term in a polynomial as an individual member
06:15	of the group , even if that term has multiplication
06:18	going on inside of it , which is common since
06:20	there's often a variable part and a number of part
06:22	being multiplied together , getting back to our example ,
06:25	distributing the factor to to each term gives us two
06:29	times three X plus two times five Y . But
06:33	this can be simplified even further because we know that
06:35	two times three is just six and two times five
06:38	is just 10 . So the distributed form is six
06:41	X plus 10 Y . Let's try another example of
06:45	a factor times of polynomial . Four times the group
06:48	X squared plus three x minus five . First .
06:52	We need to identify the terms of this polynomial .
06:54	So when we distribute the factor , we just make
06:57	one copy of it for each term . This polynomial
06:59	has three terms X squared , positive three X and
07:03	negative five . So we distribute a copy of the
07:06	factor four to each term and we get four times
07:09	X squared or just four X squared four times three
07:12	X , which is 12 X . Since four times
07:15	three is 12 and four times the negative five which
07:18	is negative 20 . So the equivalent distributive form is
07:22	four X squared plus 12 X minus 20 . Let's
07:26	see another example . X times the group X squared
07:30	minus eight X plus two . In this expression ,
07:33	the factor being multiplied by the group is actually a
07:36	variable , but the distributive property works exactly the same
07:39	way and it says that we can distribute that factor
07:42	and multiply it by each term of the group individually
07:46	. The first term is X squared , which is
07:48	the same as X times X . So if you
07:50	multiply that by X will get X cubed since that
07:52	would be three , X's multiplied together . The next
07:56	term is negative eight X . So if you multiply
07:58	that by X will have negative eight times X times
08:01	X , which is the same as negative eight X
08:04	squared . Last of all , we have the term
08:07	positive two and X times positive two is just two
08:10	X . So after distributing the factor X to each
08:13	member of the original group , we have the polynomial
08:17	X cubed minus eight X squared plus two X .
08:21	See why the distributive property is so handy in algebra
08:24	. It shows us how to multiply a polynomial by
08:27	a factor . We just distribute a copy of that
08:29	factor to each of the polynomial terms . So I
08:33	know what you're thinking . If we can distribute something
08:35	to each member of a group , can we do
08:39	the process in reverse and un distribute something ? We
08:44	sure can take a look at this polynomial four X
08:48	cubed plus four X squared plus four X . Notice
08:51	that each term of this polynomial has a factor of
08:54	four as its number part . In fact , it
08:57	kind of looks like someone distributed a factor of four
08:59	to each term . Since distributing a factor means making
09:03	multiple copies of it for each member of a group
09:05	undistributed is going to mean consolidating multiple copies of a
09:09	factor into a single copy that's multiplied by the whole
09:13	group . So in this case we can remove the
09:16	factor of four that's being multiplied by each term individually
09:19	. And then we can consolidate those into a single
09:22	factor of four that's being multiplied by the entire polynomial
09:25	by using parentheses to turn the polynomial into a group
09:29	. But mathematicians usually don't call this un distributing afore
09:32	instead they would say that we factored out of four
09:35	from the polynomial . So you can use the distributive
09:38	property both ways . If you get the expression eight
09:41	times the group B plus C . You can distribute
09:44	a copy of the factor A . To each member
09:46	of the group . But if you're given the expression
09:49	A . B plus A . C . You can
09:51	apply the distributive property and reverse and factor out the
09:54	A . So that it's multiplied by the whole group
09:56	at once . It's important to realize that neither of
10:00	these changes the value of the expression , distributing and
10:03	un distributing a factor are just ways of going back
10:06	and forth between two equivalent forms of an expression .
10:09	And it works in cases where it's not quite so
10:11	obvious to for example , look at this polynomial eight
10:15	X plus six , Y plus four . Z .
10:18	Notice that each of the number of parts of this
10:20	polynomial is an even number , which means it contains
10:24	a factor of two . Eight is two times four
10:27	. Sixes two times three and four is two times
10:30	two . So each of these terms has a common
10:33	factor of two . And that means that if we
10:35	want to we can factor out that too . We
10:38	can apply the distributive property . In reverse , we
10:41	remove the two from each term and consolidate it to
10:45	form a single factor that's multiplied by the whole polynomial
10:48	at once . And it works exactly the same way
10:51	for variables to what if we have the polynomial A
10:54	. X squared plus A X plus A . Each
10:58	of these terms has the common factor A . So
11:01	you could undistributed or factor out the A . Notice
11:04	that when we do that to the last term ,
11:06	which was just a that that term becomes a one
11:09	because there is always a factor of one being multiplied
11:11	by any term . All right . So that's the
11:15	basics of how the distributive property works in algebra .
11:18	As you can see , it can get pretty complicated
11:20	for big polynomial , but the most important thing is
11:23	to understand how it works in simple cases . So
11:26	you can build on that understanding in the future .
11:28	Being able to recognize the pattern of the distributive property
11:32	and to apply it in both directions will allow you
11:35	to rearrange algebraic expressions and equations when you need to
11:39	and remember . The key to really understanding math is
11:42	to try working some practice problems so that you actually
11:44	use what you've learned in this video . As always
11:47	. Thanks for watching Math Antics and I'll see you
11:49	next time learn more at Math Antics dot com .

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