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CCSS.Math: , ,

so we have a quadratic expression here x squared minus 3x minus 10 and what I'd like to do in this video is I'd like to factor it as the product of two binomials or to put it another way I want to write it as the product X plus a that's one binomial times X plus B where we need to figure out what a and B are going to be so I encourage you to pause the video and see if you can figure out what a and B need to be can we rewrite this expression as a product of two binomials where we know what a and B are so let's work through this together now and I'll I'll highlight a and B in different colors so I'll put a and yellow and I'll put B in magenta so if you one way to think about it so let's just multiply these two binomials using a and B and we've done this in previous videos you might want to review multiplying binomials if any of this looks strange to you but if you were to multiply what we have on the right hand side out it would be equal to you're going to have the x times the X which is going to be x squared then you are going to have the a times the X which is a X and then you're going to have the B times the X which is B X actually let me just I'm not going to skip any steps here just to see at this time this is all review or it should be review so then we have so we did x times X to get x squared then we have a times X to get a X to get a X and then we're going to have x times B so we're multiplying each term times every other term so now we have x times B to get B X so plus BX B X and then finally we have plus the a times the B which is of course going to be a a B and now we can simplify this and you might have been able to go straight to this if you are familiar with multiplying binomials this would be x squared plus we can add these two coefficients because they're both on the first degree terms they're both multiplied by X if I have a X's and I add B X's to that I'm going to have a plus B X's so let me write that down a plus B X's and then finally I have the plus do that blue color finally I have it as plus a B plus a B and now we can use this and now we can use this to think about what a and B need to be if we do a little bit of pattern matching we see we have an x squared there we have an x squared there we have something we have something times X in this case it's a negative 3 times X and here we have something times X so one way to think about it is that a plus B needs to be equal to negative 3 the they need to add up to be this coefficient so let me write that down so we have a plus B a plus B needs to be equal to negative three needs to be equal to negative three and we're not done yet we finally look at this last term we have a times B well a times B needs to be equal to negative 10 so let's write that down so we have a times B needs to be equal to needs to be equal to negative 10 and in general whenever you're factoring something a quadratic expression that has a 1 on the on the quadratic on the second degree term so it has a 1 coefficient on x squared or you don't even see it but it's implicitly there you could write this as 1 x squared way to factor is say well can I come up with two numbers that add up to the coefficient on the first degree term so two numbers that add up to negative three and if I multiply those same two numbers I'm going to get negative 10 so two numbers that add up to negative 3 to add up to the coefficient here and I want to multiply it I get the constant term I get this right over here two numbers when I multiply I get negative 10 well what could those numbers be well since since when you multiply them we get a negative number we know that they're going to have different signs and so let's see how we could think about it and since when we add them we get a negative number we know that the negative number must be the larger one so if I were to just factor 10 10 I could 10 you could view that as 1 times 10 1 times 10 or 2 times 5 and 2 & 5 are interesting because if one of the our negative their difference is three so if one is negative so let's see if we're talking about negative 10 negative 10 you could say negative 2 times 5 and when you multiply them you do get negative 10 but if you add these two you're going to get positive 3 but what if you went positive 2 times negative 5 now this is interesting because still when you multiply them you get negative 10 and when you add them two plus negative 5 is going to be negative 3 so we have just figured out we have just figured out our two numbers we could say that a and we could we could say that a is 2 or we could say that B is 2 but I'll just say that a is 2 so a is equal to 2 and B is equal to negative 5 B is equal to negative 5 and so our original expression we can rewrite as so we can rewrite x squared minus 3x minus 10 we can say that that is going to be equal to X plus 2 X plus 2 x times X instead of saying plus negative 5 which we could say we could just say so you let me write that out I could write just plus negative 5 right over there because that's our B I could just write X minus X minus 5 and we're done we've just factored it as a product of two binomials now I did it fairly involved mainly so you see where all this came from but in the future whenever you see a quadratic expression and you have a 1 coefficient on the second degree term right over here you could say all right well I need to figure out two numbers that add up to the coefficient on the first degree term on the X term and those same two numbers when I multiply I need to be equal to this constant term need to be equal to negative 10 you say okay well let's see they're going to be different signs because when I multiply them I get a negative number the the negative one's going to be the larger one since when I add them I got a negative number so let's see oh say five and two seem interesting well negative 5 and positive 2 when you add them you're going to get negative 3 when you multiply them you get negative 10