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Current time:0:00Total duration:10:36

Video transcript

and I want to do a bunch of examples dealing with probably the two most typical types of polynomial multiplication that you'll see definitely in algebra and the first is just squaring a binomial so if I have X plus 9 squared I know that your temptation is going to say oh isn't that x squared plus 9 squared and I'll say no it isn't you have to resist every temptation on the planet to do this it is not x squared plus 9 squared remember X plus 9 squared this is equal to X plus 9 times X plus 9 this is a multiplication of this binomial times itself you always need to remember that it's very tempting to think that it's just x squared plus 9 squared but no you have to expand it out and now that we've expanded it out we can use some of the skills we learned in the last video to actually multiply it and just to show you that we can do it in the way that we multiplied the trinomial last time let's multiply X plus 9 times X plus a magenta 9 and I'm doing it this way just to show you what I'm multiplying by this 9 versus this X but let's just do it so we go to 9 times 9 is 81 put it in the constants place 9 times X is 9x then we have go switch to this X term we have a yellow X x times 9 is 9x put in the first-degree space x times X is x squared x squared and then we add everything up and we get x squared plus 18x plus 81 so this is equal to x squared plus 18x plus 81 now you might see a little bit of a pattern here and I'll actually make the pattern explicit in a second but when you square binomial what happened you have x squared you have this x times this X gives you x squared you have the 9 times the 9 which is 81 and then you have this term here which is 18x how did we get that 18 X well we multiply this x times 9 to get 9x and then we multiplied this 9 times X to get another 9x and then we added the 2 right here to get 18x so in general whenever you have a squared binomial let me do it this way I'll do it in very general terms let's say we have a plus B squared a plus B squared let me multiply it this way again just to give you the hang of it this is equal to a plus B times a plus I'll do a green B right there so we have B times B is B squared and let's just say assume that this is a constant term I'll put it in the B squared right there if this is a I'm assuming this is constant so this would be a constant this would be analogous to our 81 a is the variable that we actually let me change that up even better let me make this into X plus B X plus B squared and we're assuming B is a constant so it would be X plus B times X plus a green B right there so swimming B is a constant B times B is V squared B times X is B X and then we'll do the magenta X X times B is B X and then x times X is x squared so when you add everything you're left with x squared plus 2 B 2 B X plus B squared so what you see is you have your the end product what you have when you have X plus B squared is x squared plus 2 times the product of X and B plus B squared so given that pattern let's do a bunch more of these a bunch more of these so let's and I'm going to do it the fast way so 3x minus 7 squared let's just remember what I told you and just you know just don't remember you should always have a you know in the back of your mind you should know why it makes sense if I were to multiply this out do the distributive property twice you know you'll get the same answer so this is going to be equal to 3x squared 3x squared plus 2 times 3x times negative 7 right we know that it's 2 times each the product of these terms plus negative 7 squared and if we use our product rules here 3x squared is the same thing as 9x squared this right here you're going to have a 2 times the 3 which is 6 times the negative 7 which is negative 42 X and that a negative 7 squared is plus 49 that was a fast way and just to make sure that I'm not doing something bizarre let me do it the slow way for you 3x minus 7 times 3x minus 7 7 negative 7 times negative 7 is positive 49 negative 7 times 3x is negative 21x 3x times negative 7 is negative 21 X 3x times 3x is 9x squared you scroll to the left a little bit add everything you're left with 9 x squared minus 42 X + 49 so we did indeed get the same answer let's do one more and we'll do it the fast way so if we have 8x minus 3 actually let me do it one with has more variables in it let's say we had 4x squared plus y squared and we wanted to square that well same idea this is going to be equal to this term squared 4x squared squared plus 2 times the product of both terms 2 times 4x squared times y squared plus y squared this term squared and what's this going to be equal to this is going to be equal to 16 all right 4 square root of 16x squared squared that's 2 times 2 so it's X to the fourth power and then plus 2 times 4 times 1 that's 8x squared Y squared and then Y squared squared is y to the fourth now we've been dealing with squaring a binomial the next example I want to show you is when I take the product of a sum and a difference and this one actually comes out pretty neat so I'm going to do a very general one for you let's just do a plus B a plus B times a minus B so what's this going to be equal to this is going to be equal to a times a a times a so I'm going to let me make these actually in different colors so a minus B just like that so it's going to be this green a times this magenta a a times a plus or maybe I should say minus the green a minus the green a times this B I got the minus from right there and then we're going to have the green B so plus the green B times the magenta a I'm just multiplying every term by every term times the magenta a and then finally - the green B minus the Green B that's where the minus is coming from - the green B times the magenta B times the magenta B and what is this going to be equal to this is going to be equal to a squared and then this is minus a B this could be rewritten as plus a B and then we have minus B squared these right here cancel out - a B plus a B so you're just left with a squared minus B squared which is a really neat result because it really simplifies things so let's use that notion to do some multiplication so if we say 2x minus 1 times 2 plus one well these are the same things the 2x plus one you could view this as if you like a plus B and the 2x minus one you could view it as a minus B where this is a and that B is 1 this is B this is B that is a just using this pattern that we figured out just now so what is this going to be equal to its going to be a squared it's going to be 2x squared minus B squared minus 1 squared 2x squared is 4x squared 1 squared is just 1 so minus 1 so it's going to be 4x squared minus 1 let's do one more of these just to really hit the point home so if I well actually I'll just focus on multiplication right now if I have 5a minus 2b and I'm multiplying that times 5a plus 2b and remember this only applies to when I have a product of a sum and a difference that's the only time that I can use this and I've shown you why and if you're ever in doubt just multiply it out it'll take you a little bit longer and you'll see the terms cancelling out you can't do this for just any binomial multiplication you saw that earlier in the video when we were multiplying we were taking squares so this is going to be using the pattern it's going to be 5a squared minus 2b squared which is equal to 25 a squared minus 4b squared and well I'll leave it there and I'll see you in the next video