 This video is part one of factoring. To begin with we want to use a process called x factoring and to do that we need to look at this x here. In the top of the x we're going to put a product and in the bottom of the x we're going to put a sum and then on either side is where we're actually going to find our factors and it works this way that with factors m times n would multiply to get this product and m plus n would add to get the sum so you're going to just kind of take the two numbers and do the top and the bottom and see what happens. So let's look at this one. We won't normally do it this one but I want to get you in the habit of looking at products and sums. So the product is negative five times eight or four negative forty and the sum is negative five plus eight which would be positive three and the next one negative four times negative eight would be a positive thirty two and if we take four negative four plus a negative eight we get a negative twelve. So now let's go the other way. We want to find that m and that n. So factors of eighteen that add up to eleven well you could sit here and start listing your factors one and eighteen if I add those that would be nineteen and two times nine would be two plus nine would give me eleven. So that must be my factors so I have a positive two and a positive nine again two times nine is eighteen two plus nine is eleven let's try with negative twelve if we start listing our factors we could have negative one times twelve but that would be negative eleven we're trying to get to negative one. So then we could try negative two times six that's also negative twelve that's a positive six and that would give me a positive four that doesn't work. We also have negative three times four and that would give me a positive one so I'm close here this would give me a positive one if I change my signs and make that three and negative four when I add those I will get negative one so I want negative four and positive three. So how do we use that? Well first before we really start factoring we have one more thing that we need to look at before we can actually factor. Just to notice a couple of things that are always true about factoring. So here we have x squared plus nine x plus fourteen and I've told you that it's x plus two and x plus seven. Same thing here we've got this x squared minus four x minus twelve and it factors to x plus two x minus six and again we have x squared plus two x minus three is equal to x minus one x plus three and I really should have a fourth one in here so let me give you that one and it would be x squared minus five x plus six and that one would factor to x minus three and x minus two. So if you look at the factors of the first term if you look inside our parentheses of our binomials the factors of x squared are actually these first terms so the factors of the first term are the first terms of our binomial and if you look at the last terms fourteen factors of fourteen one factor is set is two and seven. Look here negative twelve one factor set is two and negative six and also here negative three we could have negative one and three that's one way to get to negative three so the factors of the last term were the last terms of my polynomial or my binomials. Now let's look at the positive last term that would be this case here and the one that I shared with you and added on. If I have a positive last term notice I had a positive and a positive and I have a positive last term in this case I have a negative and a negative or remember to get a positive you have to multiply two of the same sign. How are we going to know whether it's a positive or a negative? Well look at your middle term. My middle term is positive nine and so are my two signs in my binomials. This is a negative five and so are the signs both negatives inside my binomials. So the positive last term the signs are the same and let's take it one step further and say they're the same as the middle. So our last two cases in it when we have a negative last term and remember to get a negative you have to multiply a negative times a positive you have to have opposite signs so you have to find the right combination but they will always be opposite. We're going to be factoring and look here we're going to make sure that A is always one in this video because things work out nicely. So if I look at this I want to say what C and B are actually we should call it AC. If I did A times C one times six would be six and my B is a positive seven and this is a positive six alright and here's where the X comes in. This is my A times my C so that's my six and this down here is always going to be my B that's my plus seven. So what are factors that I added together to get to seven that are factors of six well that would be six and one both positive because it's a positive number I need the same signs. So here I've given you X plus one so the other one must be X plus six. Say again let's look at AC that'll make it easier for the next video. So A is one and C is six this is my A and this is my C and this is my B. A times C would be again positive six and my B is this middle term which is negative five. So if I make an X for that one I have positive six in the top negative five in the bottom and I need to find factors that will add up to negative five. Now it's a positive first term that means my signs have to be the same and if I'm going to add together and get a negative number that means both my signs had to be negative. So I have factors of six one and six won't do that that would give me negative seven but a negative three and negative two will give me negative five when I add them. So it must be X it was a negative three and X minus two because it was a negative two. Alright again AC let's try a different color AC A is one C is negative eight so we have negative eight and our middle term is negative seven so again making an X here my AC goes in the top that's negative eight my B goes in the bottom that's negative seven and so factors of negative eight that will add up to negative seven well they have to be opposite signs this time since it's a negative and I want my bigger number to be negative because I'm going to have a negative result and so I think it's negative eight and positive one which adds up to negative seven. So I have X minus here for you so the eight would go in this factor and then it would be X plus one for our second factor and finally again we're going to do A times C so one times negative twelve is negative twelve B is the positive four in the middle and if we make an X for that one negative twelve and a positive four remember now we have a negative in our product which means we have to have a negative and a positive number and factors of negative twelve that will add up to four you can't think of it right away again try negative one times twelve nope that doesn't do it that's negative eleven negative two times six oh that's a positive six and there it is right there four six minus two would be positive four so we have negative two and positive six until we would say X minus two and X plus six now I want you to notice that if I gave you the option of what you wanted to put in there we could have written X plus six and X minus two these two things are the same so it doesn't matter what order you put them in you just need to make sure that one goes in each factor so let's try one from scratch so always look for your greatest common factor first we haven't been doing that because we wanted to concentrate on the how to do it but now we're looking do we have a greatest common factor well we don't this time so there's nothing there okay and my A times my C is going to be one times negative ten so that's negative ten my B is positive three so in my X I'm going to rewrite it over here because that one's got words all over it my AC goes on the top that's negative ten and my B goes in the bottom that's positive three so factors of negative ten what does that tell you it tells you that you've got a negative sign and a positive sign so we're really going to kind of subtract these numbers that will give us a result of three well negative five times two would be negative three but negative two times positive five would give us a positive three so negative two is one of our factors five is the other one and if we fill it in here we have X plus so it would be the plus five and X minus that would be our two and that would be our factorization