WEBVTT mathematics/algebra-1/smith
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Welcome back to www.educator.com.
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In this lesson we are going to take care of multiplying polynomials.
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Specifically we will look at the multiplication process in general, so you can apply that to many different situations.
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We will look at some more specific things like how you multiply a monomial by any type of polynomial
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and how can you multiply two binomials together.
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I will also show you some special techniques like how you can organize all of this information into a nice handy table.
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I will show you the nice way that you can multiply two binomials using the method of foil.
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Watch for all of these things to play a part.
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Now in order to multiply two polynomials together, what you are trying to make sure is that every term in the first polynomial gets multiplied
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by every single term in the second polynomial.
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That way you will know every single term gets multiplied by every other term.
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If you have one of your polynomials being a monomial, it only has one term and this looks just like the distributive property.
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Let us do one real quick so you can see that it is just the distributive property.
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We are going to take to 2x⁴ that monomial one term and multiplied by 3x² + 2x – 5.
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We will take it and multiply it by all of these terms right here.
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We will have 2x⁴ × 3x², then we will have 2x⁴ × 2x, and 2x⁴ × -5.
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Each of these needs to be simplified, but it is not so bad.
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You take the 2 and 3, multiply them together and get 6.
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And then we will use our product rule to take care of the x⁴ and x² by adding their exponents together x⁶.
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We will simply run down to all of the terms doing this one by one.
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2 × 2 =4, then we add the exponents on x⁴ and x¹ power = x⁵.
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At the very end, 2 × -5 = -10 and we will just keep the x⁴.
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This represents our final polynomial after the two of them multiplied together.
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Remember that we are looking so that every term in one is multiplied by every term in the other one.
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What makes this a little bit more difficult is, of course, when you have more terms in your polynomials.
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As long as you make sure that every term in one gets multiplied by every term in the other one should work out just fine.
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Be very careful with this one and see how that turns out.
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First, I’m going to take this first m³ term and make sure it gets multiplied by all three of my other terms.
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Let us put that out here.
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I need to make sure m³ gets multiplied by 2m².
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Then I will have m³ × 4m and m³ × 3.
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Now if I stop to there I would not quite have the entire multiplication process down.
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We also want to take the -2m and multiply that by all 3.
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Let us go ahead and put that in there as well.
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We will take -2m × 2m², then we will take another -2m × 4m and then -2m × 3.
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That is quite a bit but almost there.
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Now you have to take the 1 and multiplied by all 3 as well.
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1 × 2m³, 1 × 4m,1 × 3.
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That is a lot of work and we still have lots of simplifying to do,
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but we made sure that everything got multiplied so now it is just matter of simplifying.
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Let us take it bit by bit.
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Starting way up here at the beginning I have m³ × 2m², adding exponents that would be 2m⁵.
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Now I have m³ × 4m so add those exponents, and you will get 4m⁴.
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Onto m³ × 3 = 3m³ and now we continue down the list over here.
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-2m × 2m², well 2 × -2 = -4 then add the exponents and I will get m³, that will take care of that.
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We will go to this guy -2m × 4m = -8m².
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Now this -2m × 3 = -6m and that takes care of those.
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Onto the last where we multiplied one by everything.,
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Unfortunately, 1 × anything as itself we will have 2m², 4m and 3.
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I have all of my terms and the resulting polynomial, but it still not done yet.
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Now we have to combine our like terms.
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Let us go through and see if we can highlight all the terms that are like.
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I will start over here with 2m⁵ and it looks like that is the only m⁵, it has no other like terms to combine.
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We will go on to m⁴, let us see what do we got for that.
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I think that is the only one, so m⁴.
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3m³ looks like I have a couple of m³, I’m going to highlight those.
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I have some squares, I will highlight those.
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Let us see what else do we have in here, it looks like we have single m’s.
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There is that one and there is that one and there is a single 3 in the m.
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We can combine all these bit by bit.
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2m⁵, since it is the only one.
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4m⁴, since this the only one, let us check this after them.
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Now I have 2m³, 3 – 4=-1m³ and that will take care of those ones.
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-8m² + 2m = -6m².
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That one is done and that one is done.
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-6m + 4m = -2m done and done.
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And then we will just put our 3 in the end.
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You can see it is quite a process when your polynomials get much bigger,
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but it is possible to take every term and multiply it by every other term.
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Now watch for later on how I will show you some special techniques to keep track of all of these terms that show up.
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They will actually not be quite as bad as this one.
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One way we can deal with much larger polynomials and keep track of all of those terms that multiplied together
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is try and organize all of those terms in a useful way.
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I’m going to show you two techniques that you can actually organize all that information.
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One of them we will be using a table and another one we will look like more standard multiplication where you stack one on top of the other.
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What I'm trying to with each of these methods is ensure that every term in one polynomial gets multiplied by every term in the other polynomials.
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I’m not are changing the rule while we are doing a shortcut.
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We are just organizing information in a better way.
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No matter which method you use, make sure you do not forget to combine your like terms at the end so you can see the resulting polynomial.
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Let us give it a try.
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I want to multiply x² + 3x + 5 × x -4.
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The way I’m going to do this is first I’m going to write the first polynomial right on top of the second polynomial.
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From there I’m going to start multiplying them term by term and I'm starting with that -4 in the bottom,
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now multiply it by all the terms in that top polynomial.
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Let us give it a try.
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First I will do -4 × 5 = -20 then I will take a -4 × 3x = -12x and I have -4 × x² = -4x².
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That takes care of that -4 and make sure that it gets multiplied by all of the other terms.
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We will do the same process with the x.
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We will take it and we will multiply it by everything in that top polynomial.
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x × 5 = 5x and I’m going to write that one right underneath the other x terms.
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This will help me combine my like terms later.
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x × 3x = 3x².
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And one more x × x² = x³.
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I have all of my terms it is a matter of adding them up and I will do it column by column.
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This will ensure I get all of my like terms -20 - 7x - 1x² and at the very beginning x³.
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That is my resulting polynomial.
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Now another favorite way that I like to combine the terms of my polynomial is to use a table structure.
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Watch how I set this one up.
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First, along the top part of my table I'm going to write the terms of the first polynomial.
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My terms are x², 3x and 5.
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Along the side of it I will write the terms of the other polynomials, so x, -4.
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Now comes the fun part, we are going to fill in the boxes of this table by multiplying a row by a column.
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In this first one we will take an x × x².
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It feels like you are a completing some sort of word puzzle or something, only guesses would be a math puzzle.
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Also x × x² = x³.
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x × 3x = 3x² and x × 5 = 5x.
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It looks pretty good.
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I will take the next row and do the same thing.
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-4 × x² = -4x², - 4 × 3x = -12x and -4 × 5 = -20.
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You will get exactly the same terms that you do know using the other method in a different way of looking at them.
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We need to go through and start combining our like terms.
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Looking at my x³ that is my only x³ so I will just write it all by itself.
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But I have a couple of x²'s so I will write both of those and combine them together, -4x² + 3x =-x².
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Here I have -12x + 5x =- 7x and of course the last one -20.
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Oftentimes you will find your like terms are diagonals from each other, but it is not always the case that seems to be very common.
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A good important thing to recognize in the very end is that you get the same answer either way.
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Use whichever method works the best for you, and that you are more comfortable with.
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Now that we have some good methods and above, let us try multiplying these polynomials again and see how it is a little bit easier.
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I will use my table method and we will take the terms of one polynomial write along the top.
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I will take the terms of the second polynomial and write them alongside.
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You will see this will go much quicker m³ – 2m and 1, 2m², 4m and 3.
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Let us fill in the boxes.
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2m² + m³ = 2m⁵, 2 × -2 =-4m³, 1 × 2m² = 2m².
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On to the next row, 4m⁴, 4 × -2 = -8m², 4m × 1 = 4m.
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Last row, 3 × m³ = 3m³, 3 × -2 =-6m and 3 × 1 =3.
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Let us go through and start combining everything.
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I have a 2m⁵ I will write that as our first term, 2m⁵.
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I’m onto my 4m⁴ and I think that is the only one I have floating around in there, 4m⁴.
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We can call that one done.
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3m³ – 4m³, two of those I need to combine, that will be -1m³.
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I’m onto my squares, -8m² + 2m² = -6m², -6m + 4m =-2m and the last number, 3.
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The great part is that it goes through and combines all of your like terms and I know I got them off because they are all circle.
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I’m going to fix this -1, so it is just a - m³ but other than that I will say that this is a good result right here.
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Some other nice techniques you can use to multiply polynomials together is if both of those polynomials happen to be binomials.
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Remember that they have exactly two terms, this method is known as the method of foil.
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That stands for a nice little saying it tells you to multiply the first terms together, the outside terms, the inside terms and the last terms.
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It is a great way of helping you memorize and get all of those terms combined like they should.
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It also saves you from creating a large structure like a table when you do not have to.
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Let us see how it works with this one.
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I have x -2 × x - 6 I’m going to take this bit by bit.
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The first terms in each of these binomials would be the x and the other x.
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Let us multiply those together and that would give us an x².
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Then we will move on to the outside terms.
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By outside that would be the x and -6 we will multiply those together, - 6x.
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Continuing on, we are on inside terms, -2 and the x, they need to multiply together -2x.
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And then our last terms -2 × -6 = 12.
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We do get all of our terms by remembering first outside and inside last.
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With this method, oftentimes your outside and inside terms will be like terms
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and you will be able to combine them, and this one is no different.
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They combined to be 8x.
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Once you have all of your terms feel free to write them out.
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This is x² - 8x – 4 + 12 and the more you use this method, it will come in handy for a factoring a little bit later on.
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Let us try out our foil method as we go through some of these examples.
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Here I want to multiply the following binomials, 5x - 6 × 2y + 3.
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First, I'm going to take the first terms together that will be the 5x and 2y, 5 × 2 = 10x × y.
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That is as far as I can put those together since they are not like terms.
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Outside terms that would be 5x and the 3 = 15x.
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Onto inside terms, -6 × 2y = 12y and the last terms -6 × 3 = -18.
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We got our first outside, inside, last and it looks like none of these are like terms.
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I will just write them as they are 10xy + 15x -12y – 18 and we will call this one done.
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Let us try another one, and in this one you will see it has few more things that we can combine.
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We are going to multiply - 4y + x and all of that will be multiple by 2y -3x.
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Starting off with our first terms let me highlight them.
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- 4y × 2y = -8 and y × y = y².
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That is the case here of our first terms.
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Now we will do our outside terms, -4 × -3 = 12 and x × y.
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Onto the inside terms, x × 2y = 2xy.
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Of course our last terms, -3x².
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Now that we have all of our terms notice how our outside and inside terms, they happen to be like terms so we will put them together.
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That will give us our final polynomial, 8y² + 14xy - 3x².
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We can say that this one is done.
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One more example and this one is a little bit larger one.
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In fact, the second polynomial in here is a trinomial so we will not be able to use the method of foil.
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That is okay, we will still be able to multiply it together,
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but I will definitely use something like a table to help me organize my information a little bit better.
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Okay, along the top of this table, let us go ahead and write our first polynomial, x – 5y.
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Then along the rows we will put our second polynomial, I have an x² – 2xy and 3y².
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Here comes the fun part, just fill in all of those blanks by multiplying a row and a column.
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x² × x =x³, x² × -5y = -5x^2y.
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Onto the next row, -2xy × x, the x’s we can put those together as an x² and the a y.
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The last part here -2xy × -5y, let us put the y’s together, -2 × -5 =10xy².
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One more row, 3y² × x =3xy².
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I have 3y² × -5y -15y³.
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We have all of our terms in there, now we need to combine the like terms.
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Let us start here on the upper corner.
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If we have any single x³ that we can put with this one.
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It look like it is all by its lonesome, we will just say x³.
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We are looking for x^2y, they must have x² and they must have y, I think I see two of them, here is one and here is that other one.
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Let us put these together, -2 + -5 = -7 and they are x^2y terms.
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Continuing on, I have an xy².
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I have two of those so let us put them together, we will take this one and we will take that one, 10 + 3= 13xy².
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That takes care of those terms.
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One more -15y³, I will put it in -15y³.
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Now I have the entire polynomial.
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Remember, at its core when you multiply polynomials you just have to make sure that every term gets multiplied by every other term.
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Use these techniques such as foil or a table to help you organize all of those terms.
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Thank you for watching www.educator.com.