hello and welcome to algebra 2 lesson 44. in this video we're going to learn about factoring trinomials with a leading coefficient that is not 1 using the reverse foil process so again for those of you who took algebra 1 you've seen this content before so it's not too challenging for you but you might just need a refresher on it so how would we go about factoring this using reverse foil if i see something like 3x squared minus 19x minus 14 and i want to reverse the foil process so i'm going to factor this into the product of two binomials and let's just write foil out for a second so it's f for first terms o for outer terms i for inner terms and l for last terms so the first thing you would think about is the fact that this times this okay would give me this now this is a pretty easy scenario because the coefficient of x squared is a prime number right 3 is only 1 times 3 or if we're really thinking it could be negative 1 times negative 3 but you don't want to put that there because that would complicate things okay and i'll show you that at the very end what would happen if you put negative 1 and negative 3 in so i just think about 3 and 1 okay 3 and 1. now i've got an x involved there because it's x squared so i would have 3x and then x okay 3x times x is 3x squared so we've got that down and the order that i put this in wouldn't matter if i put my x here and my 3x over here that's completely irrelevant the next thing okay the next thing that you would think about is the fact that the last terms so this times this will have to give me this it'll have to give me negative 14. so i would think about factors of negative 14 but it's got to be such that the outer and the inner are going to combine to give me that negative 19x so that's where this trial and error is going to come in i've got to go through all the possibilities so for negative 14 i've got to think about positive 1 times negative 14 and i've got to think about negative 1 times positive 14. i've also got to think about positive 2 and negative 7 and then i've got to think about negative 2 and positive 7. okay so more scenarios are involved when you have something that's negative because it's a negative times a positive you've got to go back and forth between the possibilities so the only thing we can really do is just go through and do trial and error so in other words i would set up some scenarios so i have 3x i have my x and let's just do another one down here and sometimes i'll write out a bunch at once and just quickly go through the possibilities so let's start with this positive 1 and negative 14. so plus 1 here minus 14 here and then you've got to reverse that so you're going to go minus 14 and plus 1. because you've got to check different outer and inner products in other words for this one i'd have 3x times negative 14 and 1 times x now 3x times negative 14 is negative 42x 1 times x is 1x negative 42x plus x is going to be negative 41x so that's not going to work this isn't a possibility so you go to this one the outer would be 3x times 1 which is 3x the inner would be negative 14 times x which is negative 14x so negative 14x plus 3x is negative 11x so that's not going to work either so you can line that out and say well positive 1 and negative 14 will not work so then you go to the other scenario so let me let me erase this and so in this scenario we have a negative one so i'll put a negative one here and here and a positive 14. so i'll put that here and here so what am i checking so 3x times 14 would be 42x and the negative 1 times x would be negative x so 42x minus x is 41x so that won't work for this one 3x times negative 1 would be negative 3x 14 times x would be 14x so 14x minus 3x would be 11x so this is gone that's no longer a possibility we can move on to the other scenarios now when we get to this scenario here again we have positive 2 and negative 7. so plus 2 negative 7 negative 7 plus 2. all right so let's check this one so 3x times negative 7 is negative 21x and then 2 times x is positive 2x and that would work negative 21x plus 2x would give me negative 19x 2 times negative 7 is negative 14. so this is your correct factorization let me erase all this i don't need to try anything else and sometimes you hit it right away other times it takes you a while but this is our correct factorization and if you want you can use foil and check it 3x times x is 3x squared [Music] then the outer 3x times negative 7 is negative 21x then the inner 2 times x is positive 2x then the last 2 times negative 7 is minus 14. if i combine like terms here in the middle i'm going to end up with 3x squared minus 19x minus 14. that's exactly what i started with right there now let me give you a little insight into why we wouldn't want to say start out with negative 3x and negative x i can factor out a negative 1 from this if i wanted to i could say this is negative 1 times and i can make this negative 3x and this minus 2. i just pull the negative 1 out i'm legally allowed to do that then times i could pull a negative 1 out from here as well so negative 1 times you'd have negative x plus 7. this is mathematically the same as this negative 1 times negative 1 is 1. so really i could write this as negative 3x minus 2 that quantity times negative x plus 7. so i could have began if i wanted to with negative 3x and negative x so when you see something positive that's leading you can do this if you want but working with negatives in the leading position makes it a little bit more complex you want to make things as easy as you can make them so it's just better to start out with a positive here and a positive here if this is positive you know you can make that work because of what i just showed you you can always factor out a negative one from each of those multiply the two negative ones together and get one one times anything is just itself and just to completely prove this to you because i know some of you're like no that doesn't seem right if you do foil on this negative 3x times negative x is what that's three x squared negative times negative is positive 3 times 1 is 3 x times x is x squared the outer negative 3x times positive 7 is negative 21x then my inner negative 2 times negative x is positive 2x then my last negative 2 times positive 7 is negative 14. so if you combine like terms in the middle again you would get 3x squared minus 19x minus 14. so again that's why i just started out with this because it's going to be the same as this okay and it's just easier to work with positives in the leading position all right let's look at another one now and we did our first one so it should go a little bit quicker so we have 2x squared plus 23x plus 63. all right so we're going to factor this into the product of two binomials now my coefficient for the leading term here is a 2. that's a prime number so again that's a very easy scenario because i know 2 is just 2 times 1. and again for the reasons that i just covered i don't want to think about negative 2 and negative 1. so i'm just going to go 2x times x and be done with that now i just need to think about 63 here what are the factors of 63 so what times what would give me 63 but then through the process of combining like terms with the outer and the inner we would want to get a 23x in the middle and for this everything's positive so i'm just going to think about positive numbers here so 1 times 63 it's not divisible by 2 it is divisible by 3 so 3 times 21 not divisible by 4 not divisible by five is divisible by seven seven times nine and that's all we're going to get so lots of possibilities here if you think about this this right here is very far apart if you think about the numbers 1 and 63 are far apart if i was multiplying 2 times 63 i'd have a really big number or if i was just multiplying 63 times 1 i'd have 63 this one isn't going to work you can eyeball that and see that you can just eliminate that right away now 3 and 21 i might be able to get that to work i just have to check the different possibilities so 2x x so 3 and 21 so i'm only going to check 3 positive 3 here and positive 21 here and because i'm only using plus signs here it's much quicker the other scenario is positive 21 here and positive 3 here so check your outer and inner 2x times 21 would be 42x 3 times x will be positive 3x this would combine to be 45x so this isn't a possibility you can line that up for this one 2x times 3 would be 6x and then 21 times x would be 21x so that would be 27x if we added so that's not a possibility either so we can line this out so now our last possibility is to use 7 and 9. so i would do 7 here and 7 here 9 here and 9 here so the outer 2x times 9 would be 18x and then 7 times x would be 7x if you add those together you get 25x that's not 23x so this is eliminated so it has to be this possibility or it would be prime so 2x times 7 would be 14x and then 9 times x would be 9x and if we combine 14x and 9x we do get 23x so we do get that so this is the big winner so let me erase everything and again you can check this through foil 2x times x is 2x squared the outer 2x times 7 would be 14x the inner 9 times x would be 9x 14x plus 9x would be 23x so plus 23x and then 9 times 7 would be 63. so you can see you got exactly what you started with 2x squared plus 23x plus 63. all right let's look at a few more i think for most of you kind of figured this out by now we're going to look at one where the leading coefficient is not a prime number it makes it more tedious not any more difficult so 40x squared minus 172x plus 112 is what we're going to try to factor obviously if you look everything's even right i've got a 0 a 2 and a 2. but if you further think about this you should know that 40 is divisible by 4 and 112 is divisible by 4. those are obvious for 172 forget about the negative for a second for 172 well 72 is divisible by 4 so i know 172 is so that means i can pull a 4 out before i even start so if i pull a 4 out i'm going to have 10x squared minus 43x plus 28. all right so i want to factor the inside of this now so we'll have four times inside of my parentheses now if i have 10x squared it is not obvious what's going to go here and here 10 is not a prime number the factors of 10 are 1 times 10 and 5 times 2 and again i know you could do negative 1 times negative 10 or negative 5 times negative 2 but we don't need to think about that again for reasons that we already covered so i've got to go through and try as different scenarios 5x times 2x or the other scenario i would have is 4 times the quantity you'd have 10x and then times 1x or just x so these are what we're going to consider here it becomes more complex because the final term is positive and the middle term is negative so what does that tell me it tells me that to get this i had to have a negative times another negative so that tells me that these positions here are going to be negative values so what are the factors of 28 considering only the negative values well i know it's 1 times 28 so that would be let me kind of just make a line here negative 1 times negative 28 i know it's 2 times 14 so negative 2 times negative 14 i know it's 4 and 7 so we think about negative 4 and negative 7 and so that's all we're going to have so let's start out with a scenario where i have negative 1 here and i have negative 28 here now one thing that i haven't told you and i talked about this in algebra one if there's not a common factor in what you're trying to factor so in other words we're trying to factor this forget about this because you say oh there's a common factor of 4. that doesn't matter we've already pulled that out if there's not a common factor here then none of the factors for this meaning neither of these binomials would have a common factor 2 and 28 have a common factor of 2. so this isn't a possibility i can try the other alternative for this which is if i have a 28 here and a 1 here so 4 times the quantity 5x minus 28 times the quantity 2x minus 1. that's a possibility but you'll see that we're going to rule this out right away because the outer 5x times negative 1 is negative 5x and the inner negative 28 times 2x is negative 56x those two are not going to combine to give me negative 43x so that is not a possibility here now let's erase that and i'm going to move on to the next scenario which is negative 2 and negative 14. now we could eliminate that right away and why can we eliminate that right away well if i put a negative 2 here and a negative 14 here i've got a common factor of 2. if i switch the order of that and i put a negative 14 here and a negative 2 here still got a common factor of 2. and it's going to be the same thing when i get down to this if i need to if i put 14 here and 2 here common factor of 2. if i put 2 here and 14 here common factor of 2. so either way you do that in any scenario this is eliminated you don't need to worry about that now i want to check this one right here in this and if that doesn't work we're going to move on to this scenario so if i put a negative 4 here and a negative 7 here common factor of 2. so that arrangement cannot work so then what i want to try is negative 4 here and negative 7 here let's see if that works so i would have 5x times negative 7 that would be negative 35x and then i would have negative 4 times 2x which would be negative 8x so it looks like that's the big winner negative 35x minus 8x is negative 43x right that's what we want right there all right so let's erase everything and we're lucky because we didn't have to go into this scenario at all all right so we found the correct factorization if you want to you can go ahead and check this you could do 5x times 2x that would give you 10x squared the outer 5x times negative 7 will be minus 35x the inner negative 4 times 2x would be minus 8x negative 35x minus 8x would be negative 43x and then the last negative 4 times negative 7 is positive 28. so that gives you this and you're multiplying this by 4 and when you multiply it by 4 4 times 10x squared will give you 40x squared 4 times negative 43x would be minus 172 and then 4 times 28 would be positive 112. so you get exactly back to this all right let's take a look at one final problem and i gave you one here that has two variables involved you've seen factoring with this before you know at this point that it's not any more challenging right you can almost just ignore the second variable work everything out and then come back to it so if i have 8x squared plus 34xy plus 36y squared the first thing you would notice is that what everything is divisible by 2. so i could pull that out before i even begin so if i pull out a 2 i'd have 4x squared plus 17 x y plus 18 y squared and so once that's done we want to factor this inside of parentheses into the product of two binomials so i think about 4x squared i know that can come from what it could come from 4x times x or it could come from 2x times 2x so we've got kind of two scenarios here and again we're not going to think about the negative version negative 2x times negative 2x or negative 4x times negative x for the reason we talked about earlier all right so let's do this one or you'd also have this one now once we've worked this part out i know that y times y gives me y squared you can just throw this in and forget about it okay we know that from working previous problems y times y is y squared when you do your outer you get x y as a variable when you do your inner you get x y as a variable so when you combine like terms you're going to have a term with x y so just put your y in there or whatever your second variable is and forget about it don't stress about having two variables super super simple all right so the next thing is to work things out you know that you're just going to focus on this 18 here and everything's positive so i'm just going to focus on positives and so i would have what for 18 1 and 18 2 and 9 and then 3 and 6. now i've got to go through many possibilities here so let's think about for a second what we can eliminate right away for 1 and 18 it would not work in here at all because i'd have to put an 18 in here somewhere and it would have a common factor of 2. so it wouldn't work in this one it might work in this one but i'd have to try this configuration here an 18 here and a one here so just think about 4 times 18 that would give us 72 and you'd have 1 times 1 which would be 1. so 72 plus 1 would be 73 so that's not going to work out in either one so you can go ahead and get rid of that all right so let's erase this next let's think about 2 and 9. so in the first one i can't use it at all again because i'd have to put a 2 somewhere i can't use a 2 here or here because i have a common factor of 2. there's no common factor in this right here that we're trying to factor i know there's one up here but again we've already pulled that out so there shouldn't be one in any of the binomials there so it's not going to work there in this one i can put a 2 here and put a 9 here i can check that scenario so 4 times 2 would be 8 9 times 1 would be 9 8 plus 9 is 17 so we have found what we need so let's erase everything let me just kind of slide this up so we can check it through foil so i'm going to put my 2 out in front 4x times x is 4x squared my outer 4x times 2y would be positive 8xy my inside 9y times x would be positive 9xy and my last 9y times 2y would be positive 18y squared again we think about those two middle terms here 8xy plus 9xy would be 17xy so let's just erase this and put plus 17 xy and if i distributed that 2 i'd be back to 8x squared plus 34xy plus 36y squared