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  • Hi, I’m Rob. Welcome to Math Antics!

  • In this lesson, were 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 timesto 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, youll 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 youre looking at this thinking, “what multiplication?”,

  • remember that multiplication is thedefaultoperation 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 asnegativeaddition.

  • 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 equivalentdistributedform 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 youre 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 thoughttermswere 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.

  • Wellthat’s what I’m here fornoticing 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?

  • AhThat’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 exampleDistributing 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’, well get ‘x-cubedsince that would be three ‘x’s multiplied together.

  • The next term is negative 8x so if we multiply that by ‘x’ well 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 youre 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 thisUN-distributing a 4”.

  • Instead they would say that wefactored 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 youre given the expression, ab + ac, you can apply the Distributive Property in reverse

  • andfactor outthe ‘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 anevennumber 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 orfactor outthe ‘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 rememberthe key to really understanding math is to try working some practice problems

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

  • As always, thanks for watching Math Antics and I’ll see ya next time.

  • Learn more at www.mathantics.com

Hi, I’m Rob. Welcome to Math Antics!

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代数学の基礎。分布的性質 - Math Antics (Algebra Basics: The Distributive Property - Math Antics)

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    Yassion Liu に公開 2021 年 01 月 14 日
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