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Video transcript

in this video I want to do a bunch of examples of factoring a second-degree polynomial which is often called a quadratic sometimes a quadratic polynomial or just a quadratic itself or quadratic expression but all it means is a second-degree polynomial so something that's going to have a something a variable raised to the second power in this case and all of the examples will do it'll be X so let's say I have the quadratic expression x squared plus 10 X plus 9 and I want to factor it into the product of two binomials how do we do that well let's just think about what happens if we were to take X plus a and multiply that by X plus B if we were to multiply these two things what happens well we have a little bit of experience doing this this will be x times X which is x squared plus X times B which is B X plus a times X so plus a times X plus a times B plus a B or if we want to add these two in the middle right here because they have the same they're both coefficients of X we could write this as x squared plus I could write it as B plus a or a plus B X plus a B so in general if we assume that this is the product of two binomials we see that this middle coefficient this middle coefficient on the x-term or you could say the first degree coefficient there that's going to be the sum of our a and B this is going to be the sum of our a and B and then the constant term is going to be the product of our a and B notice this would map to this and this would map to this and of course this is the same thing as this so can we somehow pattern match this to that is there some a and B where a plus B where a plus B is equal to n and a times B a times B is equal to 9 well let's this think about it a little bit what are the factors of 9 what are the things that a and B could be equal to and we're assuming that everything is an integer and normally when we're factoring we're especially when we're beginning to factor we're dealing with integer numbers so what are the factors of 9 there 1 3 & 9 so this could be a 3 & a 3 or it could be a 1 & a 9 now if it's a 3 & a 3 then you'll have 3 plus 3 that doesn't equal 10 but if it's a 1 & a 9 1 times 9 is 9 1 plus 9 is 10 so it does work so a a could be equal to 1 and B could be equal to 9 so we could factor this as being X plus 1 times X plus 9 and if you multiply these two out using the skills we developed in the last few videos you'll see that it is indeed x squared plus 10 X plus 9 so when you see something like this when the coefficient on the x squared term or the leading coefficient on this quadratic is a 1 you can just say all right what two numbers add up to this coefficient right here and what two numbers add up or what to number and those same two numbers when you take their product have to be equal to 9 and of course this has to be in standard form or if it's not a standard form you should put it in that form so that you can always say okay whatever is on the first degree coefficient there my a and we have to add to that whatever is my constant term my a times B the product has to be that let's do a cup several more examples I think the more examples we do the more sense this will make let's say we had x squared plus 10x plus y or D did 10x let's do a different number x squared plus 15 X plus 50 and we want to factor this well same drill same drill we have an x squared term x squared term we have a first degree term this should be this right here should be the of two numbers and then this term the constant term right here should be the product of two numbers so we need to think of two numbers that when I multiply them I get 50 and when I add them I get 15 and this is always going to this is going to be a bit of an art that you're going to develop but the more practice you do you're going to see that it will start to come naturally so what could a and B be let's think about the factors of 50 can be 1 times 50 2 times 25 C 4 doesn't go into 50 it could be 5 times 10 I think that's all of them let's try out these numbers and see if any of these add up to 15 so 1 plus 50 does not add up to 15 2 plus 25 does not add up to 15 but 5 plus 10 does add up to 15 so this could be 5 plus 10 and this could be 5 times 10 so if we were to factor this this would be this would be equal to X plus 5 times X plus 10 and multiply it out I encourage you to multiply this out and see that this is indeed x squared plus 15 X plus 2 in fact let's do it x times X x squared x times 10 plus 10 X 5 times X plus 5 X 5 times 10 plus 50 notice the 5 times 10 gave us the 50 the 5 X plus the 10 X is giving us the 15 X in between so it's x squared plus 15 X plus 50 x squared plus 15 X plus 50 let's up the stakes a little bit introduce some negative signs in here let's say I had x squared minus 11x plus 24 now it's the exact same principle I need to think of two numbers the one I add them need to be equal to negative 11 a plus B need to be equal to negative 11 and a a times B a times B need to be equal to 24 now there's something for you to think about if when I multiply both of these numbers I'm getting a positive number I'm getting a 24 that means that both of these need to be positive or both of these need to be negative that's the only way I'm going to get a positive number here now if when I add them I get a negative number if these were positive there's no way I can add two positive numbers and get a negative number so the fact that their sum is negative and the fact that their product is positive tells me that both a and B are negative a and B have to be negative remember that one can't be negative and the other one can't be positive because then this the product would be negative and they both can't be positive because then this when you add them that would get you a positive number so let's just think about what a and B can be so two negative numbers so let's think about the factors of 24 and we'll kind of have to think of the negative factors but let me see it could be 1 times 24 1 times 24 2 times 11 8 times sorry 3 times 8 or 4 times 6 now which of these when I multiply these well obviously when I multiply 1 times 24 I get 24 when I get 2 times 11 so this 2 times 12 I get 24 so we know that all of these the products are 24 but which two of these which two factors when I add them should I get 11 and then we could say let's take the negative of both of those so when you look at these 3 and 8 jump out 3 times 8 is equal to 24 3 plus 8 is equal to 11 but that doesn't quite work out right because we have a negative 11 here but what if we did negative 3 and negative 8 negative 3 times negative 8 is equal to positive 24 negative 3 minus 11 or sorry negative 3 plus negative 8 is equal to negative 11 so negative 3 and negative 8 work so if we factor this this is going to x squared minus 11x plus 24 is going to be equal to X minus 3 times X X minus 8 let's do another one like that let's do another one like that actually let's mix it up a little bit let's say I had let's say I had x squared x squared plus 5x minus 14 so here we have a different situation the product of my two numbers is negative right a times B is equal to negative 14 my product is negative that tells me that one of them is positive and one of them is negative and when I add the two a plus B I get it being equal to five so let's think about the factors of 14 and what combinations of them when I add them if one is positive and one is negative or I'm really kind of taking the difference of the two do I get five so if I take 1 and 14 I'm just going to try out things 1 and 14 negative 1 plus 14 is negative 13 1 plus negative 14 is it is negative 1 plus 14 is 13 let me write all of the combinations that I could do and eventually your brain will just hang on so you could a negative 1 plus 14 is equal to is equal to 13 and 1 plus negative 14 is equal to negative 13 so those don't work that doesn't equal 5 now what about 2 & 7 if I do negative 2 let me do this in a different color if I do negative 2 plus 7 that is equal to 5 we're done that worked and we could have tried you know 2 plus negative 7 that would be equal negative 5 so that wouldn't have worked but negative 2 plus 7 works and negative 2 times 7 is negative 14 so there we have it we know it's X minus 2 times X plus 7 that's pretty neat negative 2 times 7 is negative 14 negative 2 plus 7 is positive five let's do it let's do like let's do it several more of these just to really really get well honed the skill so let's say we have x squared minus X minus 56 so the product of the two numbers have to be minus 56 have to be negative 56 and their difference because one is going to be positive and one is going to be negative all right the difference has to be negative one and the numbers that immediately jump out in my brain and I don't know if they jump out in your brain we just learned this in the times tables 56 is eight times seven I mean there's other numbers it's also was it 28 times two it's all sorts of things but eight times seven really jumped out into my brain because they're very close to each other we need numbers that are very close to each other and one of these has to be positive and one of these has to be negative now the fact that when they're sum is negative tells me that the larger of these two should probably be negative so if we take negative eight times seven that's equal to negative 56 and then if we take negative eight negative 8 plus 7 that is equal to negative one which is exactly the coefficient right there so when i factor this this is going to be X minus 8 times X plus seven this is often one of the hardest concepts people learn in algebra because it is a bit of an art you have to look at all of the factors here play with the positive and negative signs see which of those factors when one is positive one of negative add up to the coefficient on the x-term but as you do more and more practice you'll see that it'll become a bit of second nature now let's step up the stakes a little bit more let's say we had negative x squared everything we've done so far had a positive coefficient a positive one coefficient on the x squared term let's say we had a negative x squared let's say we had a negative x squared minus 5x plus 24 how do we do this well the easiest way I can think of doing it is factor out a negative one and then it becomes just like the problems we've been doing before so this is the same thing as negative 1 times positive x squared plus 5x - 24 right I just factored a negative one out you can multiply a negative one times all of these and you'll see it becomes this or you could factor the negative one out and divide all of these by negative one and you get that right there now same game as before I need two numbers that when I take their product I get negative 24 so one will be positive one will be negative and when I take their sum or really their well you can imagine well when I take their sum it's going to be five so let's think about 24 is one and 24 let's see if this is negative one and 24 is negative 23 otherwise it would be it if it's negative one or 24 would be positive 23 if it was the other way around would be negative 23 doesn't work what about 2 and 12 well if this is negative if then - isn't it remember one of these have to be negative if the 2 is negative their sum would be 10 if the 12 is negative they're something to be negative 10 still doesn't work 3 & 8 if the 3 is negative their sum will be 5 so it works so if we pick negative 3 & 8 negative 3 & 8 work so if we use because negative 3 plus 8 is 5 negative 3 times 8 is negative 24 so this is going to be equal to can't forget that negative 1 out front and then we factor the inside negative 1 times X minus 3 that's X minus 3 times X plus 8 and if you really wanted to you could multiply the negative 1 times this you would get 3 minus X if you did or you don't have to let's do let's do one more let's do one more of these the more practice the better I think all right let's say I had negative x squared plus 18x minus 72 so once again I like to factor out the negative 1 so this is equal to negative 1 times x squared minus 18x plus 72 now we just have to think of two numbers that when I multiply them I get positive 72 so they have to be the same sign and that makes it easier in our head at least in my head when I multiply my positive 72 when I add them I get negative 18 so they're in the same sign and their sum is a negative number they both must be negative so they're both negative both negative and we could go through all of the factors of 72 but though the one that springs up and you know I maybe you think of 8 times 9 but 8 times 9 or negative 8 minus 9 or negative 8 plus negative 9 doesn't work that turns into 17 that was close let me show you that negative 9 plus negative 8 that is equal to negative 17 close but no cigar so what other ones are we have 6 and 12 that actually seems pretty good if we have negative 6 plus negative 12 that is equal to negative 18 notice it's a bit of an art you have to try the different factors here so this will become negative 1 don't want to forget that times X minus 6 times X minus 12