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 take a look at some very special factoring techniques that you want to know.
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Some of these special factoring techniques will be recognizing the difference of squares,
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a perfect square trinomial and the difference and sum of cubes.
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Here is a great pattern to pick up on that can cut out a lot of work for you.
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Recall that when we are multiplying together a few different polynomials that some very special ones came up.
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For example, if you multiply these two together x - y and x + y, the middle term end up dropping out entirely.
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Our first term would be x², our outside terms would be x, y, inside terms -xy and last terms –y².
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The only thing that will remain since these two cancel each other out are just the x² and the - y².
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In few other situations you might have had, say x + y the whole thing squared and x - y the whole thing is squared of that one.
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Since we have been working with factoring, we want to look at all 3 of these processes in the other way.
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If we can recognize that we have one of these types of polynomials we will immediately know how it should factor on the other side.
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Think of these as like some nice handy formulas that can help you cut down on the factoring process.
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Since we are looking at these in reverse, we want to look at some key features of these formulas
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to help you recognize them when looking at those polynomials.
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Let us start off.
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When you are looking at the difference of squares here is how you can pick it out.
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One, both of these terms should be squared and notice how there is no middle term.
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One last very important thing is this is the difference of squares, so you should have subtraction.
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You can find all those 3 parts then you know how this will end up factoring.
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This will be x + y and x – y.
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For your perfect square trinomials, you want the first term and the last term to be squared.
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The last term over here should be positive.
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Now comes the tricky part, you want the middle term to be twice what you get when you multiply x and y together.
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It does not matter if it is positive or negative, since you have two different formulas for those.
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But that is what it should turn out to be.
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Now if you can recognize all of those things, then you know how these will break down.
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This one will be x + y² and that one will be x - y².
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Be careful when using these quick and easy formulas, note that there is no sum of squares formula.
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It is often very tempting to look at something like x² + y² and think that it will break down, but be careful it will not.
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Be very keen on picking up these small things that we can save yourself some time when going through that factoring process.
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Let us give it a try.
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I want to go ahead and factor x² -16².
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I think this looks like the difference of squares and I will end up rewriting it a little bit so you can tell.
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This is x² - and I need the other number to be² as well, this would have to be a 4².
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I'm looking at my two terms of being like an x and 4.
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According to my formula, this will definitely factor.
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Factor into x and x then +4 - 4.
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Let us go ahead and multiply things back together to see why this is the correct factorization.
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We take our first terms, we get x², our outside would be - 4x, inside + 4x, and our last terms -16.
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You will notice that those outside and inside terms, one is positive and one is negative, other than they are the same.
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They will end up canceling each other out.
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The only thing left with is just the x² and the 16 like we should.
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You can be pretty confident that this is the proper factorization when dealing with the difference of squares.
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The other ones that we want to look at are those perfect square trinomials.
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Let us see if we can hunt this one out.
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Looking at it, I can see that my y² and I want to write the 100 as being a squared number as well.
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y² + 20y + the number 10².
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My first term there is a y being squared and my last term is 10 being squared.
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My last term is positive and that is a good sign.
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I need to check just one more thing, is my middle term twice the first term multiplied by the last term?
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2 × y × 10.
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It does not take too much work.
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But if we do multiply these together or maybe rearrange them first, you will see that we do get a 20y like we are supposed to.
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I can use my nice little shortcut formula to go ahead and factor this.
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My first terms will be y and my last terms 10.
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Notice how y + 10 and y + 10 they are exactly the same.
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We will go ahead and we write this as a single term with just a square exponent on it.
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Nice, very quick and easy.
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Let us go ahead and multiply this out to show you that it does work.
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My first terms would be y², my outside and inside terms would be exactly the same.
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My last terms, 10 × 10 would be 100.
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The fact that our outside and inside terms are exactly the same means that we will have two of them.
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That is why we need that 2 in our shortcut formula.
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That should make you pretty confident that this is the correct factorization for our perfect square trinomial.
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Let us try some examples to see if we can get better using the special factoring techniques.
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In this first one I have y² - 81and this looks like it might be the difference of squares.
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I need to write this such that I have my first number squared and the second number squared.
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9² is the only thing that will give me 1.
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I can definitely see that it is the difference of squares.
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I have y² 9² and they are being subtracted.
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This means that I can use my nice shortcut to go ahead and break this down.
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Y - 9 and y + 9.
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Let us try something that has a few more fractions in it.
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This one is x² - 16/25.
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We want to write this such that we have something squared - something squared.
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Two things that will multiply and give me x², that will be x.
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To get that 16/25 I think I we are going to have to use 4/5.
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4 × 4 =16 and 5 × 5 =25.
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Sure enough, this is the difference of squares because both of these are squared and we are taking their difference subtracted.
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Let us write down what we have.
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x - 4/5 and x + 4/5.
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One more example, this one is u² + 36.
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You can recognize this one as u² + 6 being squared.
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Be very careful, do not go anywhere beyond that because we cannot use one of our special formulas here.
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Notice how this one we are dealing with a sum and we do not have a special formula for the sum of squares.
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I’m going to write does not factor.
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Be on the watch out for that one.
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If you have the difference of squares you are in good shape.
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If you have the sum of squares then you are stocked.
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Let us try a few more that involved the difference of squares.
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This first one we have a 49x² - 25.
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So we want to think of it as something squared - something squared.
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Two numbers that will multiply and give me my 49, that must be 7.
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7x is my first one.
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Two numbers that will multiply to 25 must be 5.
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We can write how this one breaks down.
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I have 7x - 5 and 7x + 5.
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One more, here I have two square numbers and we are taking their difference.
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We need to view this as something squared - something squared.
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See 8 × 8 = 64 I will put that in there.
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8p if I square that entire thing I will get 64p².
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9q, 9 × 9 will give us 81 and q² will give us that q².
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Now we have both of our pieces and we can use our shortcut.
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8p – 9q 8p + 9q and that one is factored completely.
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When using any of the special factoring techniques you should always be on the watch out
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for some common factors that can pull out from the very beginning.
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In these next two examples we will see if we can do just that.
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This one 50 is not a square number nor is 32 but if I look at both of them, they are both divisible by 2.
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I need to take out a common 2 first.
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Let us try that common 2 and see what is left over.
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5 0÷ 2 = 25w² and 32 ÷ 2 = -16.
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This is looking much better because 25 is a square number and so is 16.
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Let us just focus on those parts and see if we can write this as the difference of squares.
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What would give us 25 must be a 5² and 16 that is a 4².
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I can see how this one will factor.
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5w - 4 and 5w + 4.
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Do not forget that 2 that we took at the very beginning, it still hanging out front during the entire process.
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Go ahead and put it in.
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One more of this difference of squares, but this one is a little bit of a change to it.
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Notice how this one is actually to the 4th power.
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Sometimes you might be able to apply these rules, but you have to start off by looking out as something squared × something squared.
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Let us see what we can do with this one.
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What square would give us a y⁴?
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This will be trickier but if you square a y² that will do it.
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It comes from our rules for exponents because we would end up multiplying the 2’s together.
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What square would give us 81? That would be 9.
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We could use our formula to break this down y² - 9 and y² + 9.
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It is very tempting to stop right there but actually you can continue this one more step.
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If you look over here you have another difference of squares.
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You have a y² and 9 can be factored out as 3².
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Use the formula one more time to break that one down.
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What square would give you a y², y and what squared would give you 9 and 3.
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This is y - 3 and y + 3.
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Now remember this one over on the other side is still there, go ahead and write it along with the rest.
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Be careful not to try and apply your formulas to that one because that one is the sum of squares and we do not have a formula for that one.
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Now that we have seen some of those, let us get into our perfect square trinomials.
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The first one I have is x² - 24x + 144.
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Let us double check to see that this is a perfect square.
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My first term is squared and my last term is squared.
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I'm looking at something like x² - 24x and what will be squared to get 144, 12².
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In order for this to work out nicely, I want to make sure that middle term comes from taking 2 multiplied by my first term x and last term 12.
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By combining all that sure enough, you will see that we do get our 24x.
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We are in pretty good shape.
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This is -24 over here and that gives me another clue on how this will break down.
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Break down into x - 12 and x - 12 which of course we just go ahead and package up into one x -12, the whole thing squared.
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It is a very handy formula.
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Let us try the next one.
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18 x³ + 84x² + 94.
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That is quite a big one.
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In order to tackle this one, we definitely want a code for any common factors.
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One thing I can say is one they are all divisible by 2 so that will help break down quite a bit.
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And everything has x in common.
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Let us take those out and see what we have left over.
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18 ÷ 2 = 9x², 84 ÷ 2 = 42x and 98 ÷ 2 = 49.
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Let us see if we can use our formula on this.
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Out front I have a 9x², on the back I have a 49.
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You want to view this or at least that first one as being like a 3x the whole thing is being squared.
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On the end that is like a 7².
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Check to see that it meshes well with your middle term.
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Can you take your first term 3x and your last term 7 and multiply by 2 to get 42.
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2 × 3 × 7 =42 and there is still an x in there.
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It matches just fine.
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Let us use our formula and break that down.
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That will be 3x + 7.
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We are using the + in here because the 42 is positive
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and the reason why this is a 3x because that is what the first term squared would have to be.
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Do not forget that 2x out front at the very beginning it is still there.
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We will just finally go ahead and condense things since the 3x +7 appears twice.
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I will write this 3x + 7 that whole thing squared with a 2x out front.
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In addition to those special formulas that we have for factoring, there is also some for the sum and difference of cubes.
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We have not done a lot of factoring with cubes so these are important in order to break these types of problems down.
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For the sum and difference of cubes, they break down using these two formulas.
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I think these are a little bit unusual because we look at what they factor into, they factor in some rather large polynomials.
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You have this x - y, which is not too big, but then over here we have x² + xy + y².
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Let me first convince you that it is how it should factor by taking those large polynomials and multiplying them together.
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I’m going to do this using one of my tables.
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I will write the terms of one polynomial along the top and let me write the other one along the side.
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I’m using the first one here.
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To fill in this box we will multiply x × x² = x³, x² × y = -x^2y.
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I have x² y then - xy² then xy² then – ý³.
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When I combine these terms I can see that all have a single x³,
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but that my x² y will end up canceling each other out since they are different in sign.
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The same thing will happen with my xy², they are different in signs so they will cancel each other out as well.
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I just have one little lonely y³ on the end, - ý³.
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You can see that if I put these back together I do get the difference of cubes.
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These two formulas will help us factor them out.
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Another thing is it can be very difficult trying to remember these formulas.
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First try and identify what is being cubed, this x and y, because you will see that they show up in your formula in that first polynomial.
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Just x and y.
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They show up in the second polynomial as well, x and y with them being multiplied in the middle.
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It even works for the sum of cubes.
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You have your x and y just as they are and you have your x, xy and y.
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On the outside of the second one, these will always be squared.
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And one last thing that will help you get these two formulas down is look at how the signs are related.
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If you are dealing with the difference of squares in the first polynomial will have exactly the same sign.
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If you have the sum of cubes in the first polynomial again will have the exact same sign.
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The next sign present is opposite of what you used originally.
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If you had a negative over here, use the opposite now it is positive.
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Same thing applies on this one.
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If you use the positive, now this was going to be negative.
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For the last sign, this will always be positive.
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Here is how you can remember what the signs will be.
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Always start off with the same one as in your original then you will have opposite signs.
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Then the last one will always be positive.
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Same, opposite, positive.
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Let us see if we can use the sum and difference of cubes to help us factor up.
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We are going to use this on 8³ - 8 and 27r³ + 8.
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The very first thing you do is see what two things are being cubed.
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I have x and then 2³ would give me my 8.
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This will factor into a smaller polynomial and a larger one.
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The values that I put in here will be an x and 2, x² x × 2 and 2².
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I have put them the same as they are, then I have my x².
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I have them multiplied together and I have my 2².
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We do cleanup this a little bit by putting some signs.
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Same, opposite, positive.
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This is almost a completely factored.
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I will just go ahead and end up rewriting this because we usually like to put our coefficients in front of our variables.
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2x rather than writing it as x times two and we should write this as 4 rather than 2².
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Now that one is factored.
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You can see they do take a little bit more work but it does get the job done.
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Let us try this other one.
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What cubed + what cubed.
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What number cubed would give us 27?
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That would have to be a 3.
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We know that 3r is our first number in there.
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What cubed could give us an 8?
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I think we saw it before, that must have been 2.
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This will breakdown into a smaller polynomial and a much larger one.
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Let us write down our numbers.
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3r and 2, 3r² 3r × 2 and 2².
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Let us go ahead and add the signs to this.
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Same, opposite and positive.
00:25:45.100 --> 00:25:47.100
Now this one is almost done.
00:25:47.100 --> 00:25:51.600
We just need to clean it up a little bit.
00:25:51.600 --> 00:25:59.600
3r + 2 3² would be 9, 9r² is here.
00:25:59.600 --> 00:26:07.100
-3 × -2 = 6r and 2² would be 4.
00:26:07.100 --> 00:26:13.700
This one is factored completely.
00:26:13.700 --> 00:26:18.400
Let us do this one more time using the sum or the difference of cubes.
00:26:18.400 --> 00:26:26.400
Since it does take a little bit of practice to figure out what pieces are being cubed and where all of those pieces need to go.
00:26:26.400 --> 00:26:35.800
We need to recognize for this first one, something cubed + something cubed.
00:26:35.800 --> 00:26:40.800
Looks like my first one, and let us see what cubed would give me 64?
00:26:40.800 --> 00:26:44.900
That would have to be 4.
00:26:44.900 --> 00:26:51.900
I’m thinking of breaking this down into one small piece and one larger piece.
00:26:51.900 --> 00:26:54.100
We will go ahead and write down what these pieces are.
00:26:54.100 --> 00:27:06.600
I have k, 4k², k × 4 and 4².
00:27:06.600 --> 00:27:11.600
Now that we have all of our pieces, let us go ahead and put in our signs.
00:27:11.600 --> 00:27:19.600
Same, opposite, and positive.
00:27:19.600 --> 00:27:24.400
Do not forget this last step where we go ahead and clean everything up.
00:27:24.400 --> 00:27:35.400
k + 4k² - 4k + 16.
00:27:35.400 --> 00:27:42.000
Another thing that you may sometimes be tempted to do is sometimes you look at the second one
00:27:42.000 --> 00:27:46.200
and seems like you should be able to factor it in some sort of way.
00:27:46.200 --> 00:27:49.100
However, this is as far as it goes.
00:27:49.100 --> 00:27:56.000
Feel free to just leave it as it is.
00:27:56.000 --> 00:27:56.900
Let us try one last one.
00:27:56.900 --> 00:28:03.200
This one is 27x³ – 64y³.
00:28:03.200 --> 00:28:10.900
Something cubed + something cubed.
00:28:10.900 --> 00:28:13.700
What cubed would give us a 27?
00:28:13.700 --> 00:28:19.800
That must be a 3 and x³ would give and x³.
00:28:19.800 --> 00:28:26.600
To get a 64 this must be a 4y.
00:28:26.600 --> 00:28:31.400
Make sure we have our negative sign in there.
00:28:31.400 --> 00:28:40.300
We have that breakdown into a smaller one and a much larger one.
00:28:40.300 --> 00:28:44.200
The pieces are 3x and 4y.
00:28:44.200 --> 00:28:48.900
Be very careful as you put in those pieces of the much larger one.
00:28:48.900 --> 00:28:55.500
Remember we have 3x², we have 3x × 4y.
00:28:55.500 --> 00:28:58.600
We have 4y all of that squared.
00:28:58.600 --> 00:29:03.900
One common mistake I see with these is many people only square just the y.
00:29:03.900 --> 00:29:06.500
But it is the entire thing that means to be squared.
00:29:06.500 --> 00:29:11.400
The 4 and y.
00:29:11.400 --> 00:29:13.400
I will put in some signs.
00:29:13.400 --> 00:29:21.500
Same, opposite, and positive.
00:29:21.500 --> 00:29:25.300
One last step, let us go ahead and clean it up.
00:29:25.300 --> 00:29:32.800
3x – 4y now I have the entire thing 3x being squared.
00:29:32.800 --> 00:29:52.800
That will be 9x² 3 × 4 would be 12, so 12 xy and 4y² is 16y².
00:29:52.800 --> 00:29:56.200
This one is factored completely.
00:29:56.200 --> 00:30:01.400
That definitely get familiar with the special formulas they can save you lots of time and works.
00:30:01.400 --> 00:30:03.400
You do not have to go through as much.
00:30:03.400 --> 00:30:06.000
Remember to look for those key patterns when using these formulas.
00:30:06.000 --> 00:30:11.900
Always remember we do not have a sum of squares formula, so watch out for that one.
00:30:11.900 --> 00:30:14.000
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