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 work on factoring trinomials.
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We are not going to tackle up all types of trinomials just yet.
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For the first part we are going to focus on ones where the squared term has a coefficient of 1.
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We will also look at polynomials where we can factor out a greatest common factor.
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In future lessons we will look at the more complicated trinomials.
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One of our first techniques we have to dig back in our brains and recall how we used foil in order to multiply two binomials together.
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For example, what did we do when we are looking at x -3 × x +1.
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Using the method of foil we would multiply our first terms together and get something like x².
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We would multiply our outside terms together 1x then we would multiply our inside terms.
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And finally we would multiply our last terms together.
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Sometimes we had to do a little bit of work to clean this up.
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As long as we made sure that everything got multiplied by everything else.
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We where assured that we could multiply these two binomials out.
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Since we are working with factoring and breaking things down into a product you want to think of this process, but do it in reverse.
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If you had a trinomial to begin with, how could you then break this down into two binomials?
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Since this is the exact same one as before I will simply write down the two binomials that it will break up into.
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This is the process that we are after of taking a trinomial and breaking it down into two binomials.
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In doing so it is not quite a straightforward process.
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The way we are going to attack this is to think of that foil process in our minds.
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This will help us determine our first and last terms in those binomials.
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Now, after we have chosen something for those first and last terms
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we will have to check to make sure that the outside and inside terms combined to be our middle term.
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Sometimes we will have to do some double checking just to make sure that it does combine and give us that middle term.
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Sometimes there are lots of different options.
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We may have to do this more than once until we find just the right values that make it work.
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Watch how that works with this trinomial.
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I have x² + 2x -8 and what we are looking to do is break this down into two binomials.
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I’m going to go ahead and write down the parentheses just to get it started.
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I first want to determine what should my first terms be in order to get that x².
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We are looking to multiply two things together and get x².
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The only two things that will work is x and x.
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We have a good chance that those are our first terms.
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Rather than worrying about the outside and inside just yet, we jump all the way to the last terms.
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That will be here and here and I'm looking to multiply them and get -8.
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Here is the thing there are lots of different options that we could have, it could be 1 and 8, 2 and 4.
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We could also look at the other order of these maybe 4 and 2, 8 and 1.
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We want to choose the proper pair that when combined together will actually give us that 2x in the middle.
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Let us go ahead and try something.
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Watch how this process works.
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Suppose I just tried the first thing on the list, this 1 and 8.
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1, 8, with this combination I can be sure that my first terms work out and that my last terms work out.
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I'm not completely confident until I check those outside and inside terms to make sure that they work out.
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I will do some quick calculations.
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Let us check our outside terms, 8 × x = 8x and inside terms 1 and x, and these would combine to give us a 9x.
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If you compare that to the original it is not the same.
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What that is indicating is that pair of 1 and 8, those are not the ones we want to use.
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We can backup a little bit and try another pair of numbers.
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Let me try something different.
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I'm going to try 4 and -2.
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Now when I do my outside and inside terms I will get -2x on the outside,
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4x on the inside and those combine to give me 2x, which is the same as my middle term.
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I know that this is how it should be factored.
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If you want you can go through the entire foil process, just to double check that all the rest of terms work out.
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In fact it is not a bad idea when you are done with the factoring process, just to make sure it is okay.
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Now, there are a few tips to help you along the way when doing this reverse foil method.
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It works good, as long as you are leading coefficient is 1 and you can take a look at the signs of the other two coefficients.
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In general, here is what you are trying to do.
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You are looking for two integers whose product will give you c.
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They multiply and give you c but whose sum is b.
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It is what we did in that last example.
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Now you can get a little bit more information if you look at the signs of b and c.
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If b and c are both positive then those two integers you are looking for must also be positive.
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One situation that might happen is you know both integers that you are looking for will be negative if c is positive,
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that is this one in the end and d is negative.
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Watch for that to happen.
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Of course one last thing, you will know that the integers you are looking for are different in sign, one positive and one negative if c is negative.
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That is the only way they could multiply together and give you a negative number here on the end.
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Watch for me to use these shortcuts here in just a little bit.
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We want to use this reverse foil method in order to factor the following polynomial y² + 12y + 20.
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I’m going to start off by writing set of parentheses this will break down into some binomials.
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Let us start off for those first terms.
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What times what would give us a y²?
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One thing that will do it is just y and y.
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Let us look for two values that would multiply and give us a 20.
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It is okay for me to write down some different possibilities like 1 and 20, 2 and 10, 4 and 5.
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You can imagine the same values just flipped around.
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Let us see if we can use any information to help us out.
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Notice how this last term out here is positive and so is my middle term, both of them are positive.
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Now what is that telling me about my signs, I know that the two numbers I'm looking for will both be positive.
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That actually is quite a bit of information because now when we look at our list I can pick two things that will add to be the middle term 12.
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I know they multiply between.
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Let us drop those in there, 2 and 10.
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I have factored the trinomial.
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Let us quickly go through the foil process just to make sure that this is the one we are looking for.
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We are looking to make sure that this matches up with the original.
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First terms would be a y², outside terms 10y, inside terms 2y and last term is 20.
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These middle ones would combined giving us y² + 12y + 20 and that shows that our factorization checks out.
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We know that this is the proper way to factor it.
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Let us try another one, this one is x² - 9x – 22.
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Let us start off in much the same way.
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Let us write down a set of parentheses and see if we can fill in the blanks.
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We need two numbers that when multiplied together will give us x².
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That must be an x and another x.
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Now we need a pair of numbers that will multiply and give us a -22.
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Let us write down some possibilities like 1 and 22, 2 and 11, I think that is it.
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Let us get some information about the signs of these numbers.
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I’m looking at this last number here and notice how it is negative,
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I know that these two numbers I’m looking for on my last terms, one of them must be positive and one of them must be negative.
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The question is which one is positive and which one is negative?
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We will look to our middle term to help out.
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I need the larger term to be negative, so that I will get a negative in the middle, that -9.
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We have plenty of information it should be pretty clear that it is actually the 2 and 11 off my list that will work.
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2 and 11.
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Let us just check it real quick by foiling things out to make sure that this is how it should factor.
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First terms x², outside terms -11x, inside terms 2x, and last terms -22.
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Since I have a positive × negative, these middle terms combined I will get x² - 9x -22.
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That is the same as my original, so I know that I have factored it correctly.
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Let us use the reverse foil method for this polynomial.
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It is not very big, it is r² + r + 2.
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We are going to start off by setting down those two parentheses.
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We are hunting for two first terms that will multiply to give us r².
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There is only one choice that will do that, just r and r.
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We turn our attention to the last terms and we need them to multiply to be 2.
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Unfortunately, there is only one possibility for that, 1 and 2.
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Let us hunt down our signs.
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The last term is positive, the middle term is positive that says both of the terms that we are looking for must both be positive.
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Let us put them in and now we can go and check this using the foil process.
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r × r =r², the outside terms should be 2r, inside terms 1r, and the last term is 2.
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Combining the two middle terms here, I get r² + 3r + 2.
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Something very interesting is happening with this one, let us take a closer look.
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If we look at the resulting polynomial that we got after foiling out and we compare that with the original, they are not the same.
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That tells us something. It tells us that this is not the correct factorization.
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Now if this is not how it should be factored then what other possibilities do we have?
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If we look at all of our possibilities for those last terms, it must contain 1 and 2 if it is going to multiply and give us 2.
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Since that did not work and I have no other possibilities, it tells us that this does not factor into two binomials using 1 and 2.
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This is an indication that our original is actually prime.
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Watch out for ones like this where when you try and factor it, it simply does not factor into those two binomials.
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Let us try something with a few more variables in it.
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This one is t² – 6tu + 8u².
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Even though we have a few more variables in here, you will see that this process works out the same as before.
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Starting off with those first terms, something × something will give us t².
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That must be t and another t.
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I need two things to multiply and give us 8u².
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Well I'm not quite sure about that 8 yet because it could be 1 and 8, 2 and 4, or could be those reversed.
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I did know about the u, you better have a u and another u in order to get that u².
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Let us just focus on numbers for bit and see what information we can get from there.
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Looking at the sign of my last term it is positive, but my middle term is negative.
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The information I’m getting from there is that both of my numbers I’m looking for must both be negative.
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I think we can pick it out from our list now and it looks like we must use the 2 and 4.
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Finally let us check that to make sure this one works.
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Let us be careful since we have both t’s and u’s.
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First terms t², outside terms -4tu, inside terms -2tu and my last terms – 2 × -4 = 8u².
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We can combine our middle terms giving us t² – 6tu + 8u².
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And now we can see that yes, this has been factored correctly since the resulting polynomial is the same as my original.
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This one is good.
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One thing to watch out for is to make sure you pull out any common factors at the very beginning.
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That is what we will definitely need to do with this example before we even start the forming process.
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Always check out for a good common factor.
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This is also good idea because it can potentially make your number smaller, so that you do not have to think of as many possibilities.
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We are going to quickly factor 3x⁴ -15x³ + 18x².
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Look with this one, everything here is divisible by 3 and I can pull out an x² from all of the variables.
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Let us take it out of the very beginning, I have 3x² and let us write down what is left.
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3x⁴ ÷ 3x² = x², =15 ÷ 3 = -5x, 18 ÷ 3 + 6 and I think that is all my leftover parts.
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With this one now, I will go and do the reverse foil process on that.
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I need two binomials, let us break it down.
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What should my first terms be in order to get my x²?
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That must be an x and an x.
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I have to look at my last terms and then multiply together to give us a 6.
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1 and 6, possibly 2 and 3, but you can use the signs to help you out.
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The last term is positive, the middle term is negative, so both of these will be negative.
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It looks like I need to use that 2 and 3.
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Some quick checking to make sure this is the correct factorization.
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I have x² - 3x - 2x + 6 and looks like my outside and inside terms do combine and give us that - 5x.
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This is the correct factorization and let us go head and write out that very first 3x² that we took out at the very beginning.
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Now that we have all the pieces, we can say that this is the correct factorization.
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Always look for a common factor that you could pull out from the very beginning before starting the foil method.
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Sometimes you can pull something out, sometimes you can not but it will make your life easier if you can find something.
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Keep that in mind for this next example.
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This one is 2x³ - 18x² - 44x.
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Let us come at this over.
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It looks like everything is divisible by 2 and they all have an x in common.
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Let us take out a 2x at the very beginning.
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What do we have left?
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2x³ ÷ 2x = x², -18 ÷ 2 = -9x and -44x ÷ 2x = -22.
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Now we want to factor that into some binomials.
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Let us go ahead and copy over this 2x just we can keep track of it.
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I need my first terms to multiply together and get an x².
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That will be an x and another x.
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I need to look at my last terms so that they multiply together to give me a -22.
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Some of our possibilities are 1 and 22, 2 and 11.
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Since the last one is negative and my middle term is negative, I know that these will be different in sign.
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Let us take the 2 and 11 off of our list, those are the ones we need, -11 and 2.
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Let us quickly combine things together and make sure that it is the correct factorization.
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x² - 11x + 2x – 22.
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Combining these middle guys x² - 9x -22, so that definitely checks with this polynomial right here.
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Now if you want to go ahead and put in the 2x as well, this will take you back all the way to the original one.
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Remember to use your distribution property so you can see how that will work out.
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2x³ - 18x² - 44x and sure enough that is the same as the original.
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Everything checks out. I know that this is the correct factorization for our polynomial.
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Just a few things, make sure that when you are using this method, always check for common factor to pull it out.
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Make sure you set down your first terms and then your last terms
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and definitely check those signs to help you eliminate some of your possibilities.
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