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Video transcript

in this video I want to focus on a few more techniques for factoring polynomials and in particular I want to focus on quadratics that don't have a 1 as a leading coefficient for example if I wanted to factor 4x squared + 25 X minus 21 everything we factored so far all of the quadratics we've factored so far had either a 1 or a negative 1 where this 4 is sitting all of a sudden now we have this 4 here so I'm going to teach you is a technique called factoring by grouping and it's a little bit more involved than what we've learned before but it's a neat trick but in to some degree it'll become obsolete once you learn the quadratic formula because frankly the quadratic formula is a lot easier but this is how it goes I'll show you the technique and then at the end of this video I'll actually show you why it works so what we need to do here is we need to think of two numbers we're going to think of two numbers a and B where a times B is equal to 4 times negative 21 so a times B is going to be equal to 4 times negative 21 is equal to 4 times negative 21 which is equal to negative 84 and those same two numbers a and B a plus B need to be equal to 25 they need to be equal to 25 let me be very clear this is the 25 so they need to be equal to 25 this is where the 4 is so they equal equal 4 times negative 21 that's a negative 21 so what two numbers are there that would do this well we have to look at the factors of negative 84 and once again one of these are going to have to be positive the other one's going to have to be negative because their product is negative so let's think about the different factors that might work 4 and negative 21 look tantalizing but when you add them you get negative 17 or if you had negative 4 and 21 you'd get positive 17 doesn't work let's try some other combinations 1 and 84 too far apart when you take their difference because that's essentially what you're going to do if one is negative and one is positive too far apart let's see you could do three notes see I'm skipping something the gun two and forty two once again too far apart negative two plus 42 is 42 plus negative two is negative 42 far apart three and C 3 goes into 84 3 goes into let me just three goes into 80 4 goes into 8 two times two times three is six eight minus six is to bring down the four goes exactly eight times so three and twenty eight this seems interesting three and twenty eight we have if and wherever one of these has to be negative so if we have negative three plus twenty eight that is equal to 25 now we found our two numbers but it's not going to be quite as simple of an operation is what we did when this wasn't a a one or or when this wasn't or when this was a one or a negative one what we're going to do now is split up this term right here we're going to split this up into into negative let me make it very clear and split it up into positive 28 X minus three X we're just going to split that term and that term is that term right there and of course you have your minus 21 there and you have your four x squared over here now you might say how did you pick the 28 to go here and the three negative three to go there and it actually does matter the way I thought about it is 3 or negative 3 and 21 or negative 21 they have some common factors in particular they have the factor 3 in common and 28 and 4 have some common factors so I kind of grouped the 28 on the side of the 4 and you're going to see what I mean in a second if we literally group these so that term becomes 4x squared + 28 X and then this side over here this side over here in pink well I could let's say it's plus negative 3 X minus 21 once again I picked these I grouped the negative three with the 21 or the negative 21 because they're both divisible by three and I group the 28 with the four because they're both divisible by four and now in each of these groups we factor as much out as we can so both of these terms are divisible by four X so this orange term is equal to four x times X is 4x squared divided by four X is just X plus 28 X divided by four X is just seven now the second term remember you factor out everything that you can factor out well both of these terms are divisible by three so let's or negative three so let's factor out a negative three and this becomes X plus seven and now something might pop out at you we have X plus seven times four X we have X plus seven times four X plus X plus seven times negative three so we can factor out an X plus seven we can factor out an X plus seven this might not be completely obvious you're probably not used to factoring out an entire binomial but you could view this this could be like a or you know if you have 4x a minus three a you would be able to factor out an A and I could just leave this as a minus sign let me delete that let me delete let me delete this plus right here so K is just minus three it's just minus three right plus negative three same thing as minus three so what can we do here we have X plus seven times four X we have an X plus seven times negative three let's factor out the X plus seven we get X plus seven times 4x 4x minus 3 minus minus that three right there and we've factored our binomial we find we sorry we factored our quadratic by grouping and we factored it into two binomials let's do another example of that because it's a little bit involved but once you get the hang of it it's kind of its kind of fun so let's say we want to factor 6x squared plus seven x plus one same drill we want to find a times B that is equal to one times 6 that is equal to one times six which is equal to six and we want to find an A plus B needs to be equal to seven this is a little bit more straightforward what what are the well but the obvious one is 1 and 6 right 1 times 6 is 6 1 plus 6 is 7 so we have a is equal to 1 or let me not even assign them the numbers here are 1 & 6 now we want to split this into a 1 X and a 6 X but we want to group it so it's on the side of something that it shares a factor with so we're going to have a 6x squared here plus and so I'm going to put the 6x first because 6 and 6 share a factor and then we're going to have plus 1 X right 6x plus 1 X is 7x that was the whole point they had to add up to 7 and then we have the final plus 1 there now in each of these groups we can factor out as much as we like so in this first group let's factor out a 6 X so this first group becomes 6x times 6x squared divided by 6x is just an X 6x divided by 6x is just a 1 and then in the second group we're going to have well we're going to have a plus here but this second group we just literally have a X plus 1 or we could even write a 1 times an X plus 1 you can imagine I just factored out a 1 so to speak now I have 6x times X plus 1 plus 1 times X plus 1 well I can factor out the X plus 1 if i factor out an X plus 1 that's equal to X plus 1 times 6x plus that one I'm just doing the distributive property in Reverse so hopefully you didn't find that too bad and now I'm going to actually explain why this little magical system actually works why it actually works let me let me take an example let's say I have well in very general terms let's add a X plus B times C X and actually I don't want to use well I'm afraid to use the A's and the B's I think that'll confuse you because I use A's and B's here and they won't be the same thing so let's let me call it and let me use completely different letters let's say I have F X plus G times H X plus I'll use J instead of I you'll learn in the future why I don't like using I as a variable so what is this going to be equal to well it's going to be FX times HX which is f H X and then F X times J so plus F J X and then we're going to have G times H X so plus G H X and then G times J plus G J or if we add these two middle terms if you add the two middle terms you have F H times X plus add these two terms F J plus G H X plus G J now what what I do here well remember when in all of these problems where you have a non 1 or non negative 1 coefficient here we look for two numbers that add up to this whose product is equal to the product of that times that well here we have two numbers that add up F let's say that a is equal to F J let's say that a is equal to F J that is a and B is equal to G H so a plus B is going to be equal to that middle coefficient a plus B is going to be equal to that middle coefficient there and then what is a times B a times B is going to be equal to FJ x GH x GH which we could just reorder these terms we're just multiplying a bunch of terms so that could be rewritten as f times H times G a times J these are all the same things well what is f H times G J this is equal to F H times G J well this is equal to the first coefficient times the constant term so if a will a plus B will be equal to the middle coefficient and a times B will equal the first coefficient times the constant term so that's why this whole factoring by grouping even works or how we're able to figure out what a and B even are now I'm going to close up with something slightly different but just to make sure that you have a well-rounded education in factoring things and what I want to do is teach you to factor things a little bit more completely and this is a little bit of a add-on I was going to make a whole video on this but it's I think on some level might be a little obvious for you so let's say we had let's say we had to let me get a good one here let's say we had negative x to the 3rd plus 17 x squared minus 70 now 970 X now immediately said gee I this isn't even a quadratic I don't know how to solve something like this as X to the third power and the first thing you should realize is that every term here is divisible by X so let's factor out an X or even better let's factor out a negative x so if you factor out a negative x this is equal to negative x times negative x to the third divided by negative x is x squared 17 x squared divided by negative x is negative 17 X negative 70 X divided by negative x is positive 70 the X is canceled out and now you have something that might look a little bit that might look a little bit familiar we have just a standard quadratic where the leading coefficient is a 1 so we just have to find two numbers whose product is 10 or sorry whose product is 70 and that add up to negative 17 and the numbers that immediately jumped into my head are negative 10 and negative 7 you take their product you get 70 you add them up you get negative 17 so this part right here is going to be X minus 10 times X minus 7 and of course you have that leading negative X the general idea here is just see if there's anything you can factor out and then I'll get into a form that you might recognize hopefully you found this helpful I want to reiterate what I showed you at the beginning of this videos I think it's a really cool trick so to speak to be able to factor things that have a non 1 or non negative 1 leading coefficient but to some degree you're going to find out easier ways to do this especially with the quadratic formula and not too long