 ## еӯ—е№•иЎЁ еӢ•з”»гӮ’еҶҚз”ҹгҒҷгӮӢ

• Hi, IвҖҷm Rob. Welcome to Math Antics!

• In this lesson, weвҖҷre 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 вҖҳ3вҖҷ 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 arenвҖҷt 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 donвҖҷt know what value вҖҳxвҖҷ is.

• It could be вҖҳ4вҖҷ like in the last expression, but it doesnвҖҷt have to be.

• It could be ANY number at all! And since we donвҖҷt know what it is, that means we CANвҖҷT 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 вҖҳ3 timesвҖҷ to each member of the group

• so the group goes away and we end up with 3 times вҖҳxвҖҷ plus 3 times 6.

• The 3 time вҖҳxвҖҷ canвҖҷt be simplified any further because we still donвҖҷt know what вҖҳxвҖҷ 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 canвҖҷt simplify these expressions all the way down to a single numeric answer without knowing the value of вҖҳxвҖҷ,

• 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, youвҖҷll often see the Distributive Property shown like this:

• вҖҳaвҖҷ 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 вҖҳaвҖҷ is being multiplied by the entire group.

• But in the second form, the factor вҖҳaвҖҷ has been distributed so itвҖҷs being multiplied by each member of the group individually.

• And if youвҖҷre looking at this thinking, вҖңwhat multiplication?вҖқ,

• remember that multiplication is the вҖҳdefaultвҖҷ operation which is why we donвҖҷt have to show it in this pattern.

• Since the вҖҳaвҖҷ is right next to the group, it means itвҖҷs being multiplied by the group,

• and on this other side, since the copies of the вҖҳaвҖҷ are right next to the вҖҳbвҖҷ and вҖҳcвҖҷ, 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 вҖҳnegativeвҖҷ 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.

• ItвҖҷs usually just shown with two members in the group, but remember that it works for groups of any size.

• We could have вҖҳaвҖҷ times the group (b + c + d) and the equivalent вҖҳdistributedвҖҷ form would be: ab + ac + ad

• HereвҖҷs 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).

• AndвҖҰ вҖҳaвҖҷ times the group (x - y + 2) can be changed into the distributed form: ax - ay + a2 (or 2a which is more proper).

• So whether youвҖҳre 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?

• ButвҖҰ I thought вҖңtermsвҖқ 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.

• WellвҖҰ thatвҖҷs what IвҖҷm here forвҖҰ 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)вҖҰ

• the вҖҳxвҖҷ and the вҖҳyвҖҷ 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? вҖҰlike 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 вҖҳ2вҖҷ 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.

• WhatвҖҷs up with that?

• AhвҖҰ ThatвҖҷs 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 вҖҳ2вҖҷ 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 exampleвҖҰ Distributing the factor вҖҳ2вҖҷ to each term gives us вҖҳ2вҖҷ times вҖҳ3xвҖҷ plus вҖҳ2вҖҷ times вҖҳ5yвҖҷ.

• 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

• LetвҖҷs try another example of a factor times a polynomial: 4 times the group (вҖҳx squaredвҖҷ + 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: вҖҳx squaredвҖҷ, positive 3x and negative 5.

• So we distribute a copy of the factor вҖҳ4вҖҷ to each term and we get:

• 4 times вҖҳx squaredвҖҷ (or just 4 вҖҳx squaredвҖҷ)

• 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 вҖҳx squaredвҖҷ + 12x - 20

• LetвҖҷs see another example: вҖҳxвҖҷ times the group (вҖҳx squaredвҖҷ - 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 вҖҳx squaredвҖҷ (which is the same as вҖҳxвҖҷ times вҖҳxвҖҷ)

• so if we multiply that by вҖҳxвҖҷ, weвҖҷll get вҖҳx-cubedвҖҷ since that would be three вҖҳxвҖҷs multiplied together.

• The next term is negative 8x so if we multiply that by вҖҳxвҖҷ weвҖҷll have negative 8 times вҖҳxвҖҷ times вҖҳxвҖҷ

• which is the same as negative 8 вҖҳx squaredвҖҷ.

• Last of all we have the term positive 2, and вҖҳxвҖҷ times positive 2 is just 2x,

• so after distributing the factor вҖҳxвҖҷ to each member of the original group, we have the polynomial:

• вҖҳx cubedвҖҷ - 8 вҖҳx squaredвҖҷ + 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 polynomialвҖҷs terms.

• So I know what youвҖҷre thinkingвҖҰ

• if we can distribute something to each member of a groupвҖҰ

• Can we do the process in REVERSE and UN-distribute something?

• We sure can!вҖҰ

• Take a look at this polynomial: 4 вҖҳx cubedвҖҷ + 4 вҖҳx squaredвҖҷ + 4x

• Notice that each term of this polynomial has a factor of вҖҳ4вҖҷ 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 donвҖҷt call this вҖңUN-distributing a 4вҖқ.

• Instead they would say that we вҖңfactored out a 4вҖқ from the polynomial.

• So you can use the Distributive Property both ways.

• If you get the expression вҖҳaвҖҷ times the group (b + c), you can distribute a copy of the factor вҖҳaвҖҷ to each member of the group.

• But if youвҖҷre given the expression, ab + ac, you can apply the Distributive Property in reverse

• and вҖңfactor outвҖқ the вҖҳaвҖҷ so that it is multiplied by the whole group at once.

• ItвҖҷs 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 itвҖҷs 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 вҖҳevenвҖҷ number which means it contains a factor of вҖҳ2вҖҷ.

• 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 вҖҳ2вҖҷ and that means that if we want to, we can factor out that вҖҳ2вҖҷ.

• We can apply the Distributive Property in reverse!

• We remove the вҖҳ2вҖҷ from each term and consolidate it to form a single factor thatвҖҷs multiplied by the whole polynomial at once.

• And it works exactly the same way for variables too.

• What if we have the polynomial: вҖҳaвҖҷ вҖҳx squaredвҖҷ + ax + a

• Each of these terms has the common factor вҖҳaвҖҷ so you could UN-distribute or вҖңfactor outвҖқ the вҖҳaвҖҷ.

• Notice that when we do that to the last term (which was just вҖҳaвҖҷ) that term becomes a вҖҳ1вҖҷ

• because there is always a factor of вҖҳ1вҖҷ being multiplied by any term.

• Alright, so thatвҖҷs 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 rememberвҖҰ the key to really understanding math is to try working some practice problems

• so that you actually use what youвҖҷve learned in the video.

• As always, thanks for watching Math Antics and IвҖҷll see ya next time.