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Current time:0:00Total duration:9:49

Video transcript

in this video I'm just going to multiply a ton of polynomials and hopefully that'll give you enough exposure to feel confident when you have to multiply any for yourselves so let's start with a fairly simple problem let's say we just want to multiply 2x times 4x minus 5 well we just straight-up use the distributive property here and really when we do all of these polynomial multiplications all we're doing is the distributive property repeatedly but let's just do the distributive property here this is 2x times 4x this is 2x times 4x Ty plus 2x times negative 5 or we could say negative 5 times 2x so we could say minus 5 times 2x all I did is distribute the 2x this first term is going to be equal to we can multiply the coefficients remember 2x 2x times 4x is the same thing as you can rearrange the order of multiplication this is the same thing as 2 times 4 times X times X which is the same thing as 8 times x squared remember X to the 1 times X to the 1 add the exponents or we I mean you know x times X is x squared so this first term is going to be 8x squared and the second term negative 5 times 2 is negative 10 X not too bad let's do a slightly more involved one let's say we had 9 X to the third power times 3x squared minus 2x plus 7 so once again we're just going to do the distributive property here so we're going to multiply the 9 X to the 3rd times each of these terms so 9x to the third times 3x squared that's going to I'll write it out this time in the in the next few we'll start doing it a little bit in our heads so this is going to be 9x to the third times 3x squared and then we're going to have plus or let me write it this way minus 2 x times 9x to the third and then plus 7 times 9 X to the 3rd so sometimes I wrote the 9 X to the 3rd first sometimes wrote it later because I wanted this negative sign here but it doesn't make a difference on the order that you're multiplying so this first term here is going to be what 9 times 3 is 27 times X to thee we can add the exponents we learned that in our exponent properties this is X to the fifth power minus 2 times 9 is 18 X to the we have X to the 1 X to the third X to the fourth power plus 7 times 9 is 63 X to the third so we end up with this nice little fifth degree polynomial now let's do one where we are multiplying two binomials and I'll show you what I mean in a second and this you're going to see very very very frequently in algebra so let's say you have X minus 3 times X plus 2 and I actually want to show you that all we're doing here is the distributive property so let me write it like this let me write it like this times X plus 2 so let's just pretend that this is one big number here and it is you know if you knew what X is this would be some number here so let's just distribute this onto each of these variables so this is going to be X minus 3 times that Green X plus plus X minus 3 X minus 3 times that green to all we did is distribute the X minus 3 this is just the distributive property remember if I had a times a times X plus 2 what would this be equal to this would be equal to this would be equal to a times X plus a plus a times plus a times 2 so over here you could just you an X minus 3 is the same thing as a we're just distributing it and now we would do the distributive property again in this case we're distributing the X now on to the X minus 3 we're going to distribute the 2 on to the X minus 3 you might be used to seeing the X on the other side but either way we're just multiplying it so this is going to be on state colour-coded this is going to be x times X minus minus 3 times X plus plus X times 2 going through great pains to keep it color-coded for you I think it's helping minus 3 times 2 all I did is distribute the X and distribute the 2 and soon you're going to get used to this one we can do it in one step you're actually multiplying every term by every term in this one by every term in that one and we'll figure out faster ways to do it in the future but I really want to show you the idea here so what's this going to equal this is going to equal x squared this right here is going to be minus 3x this is going to be plus 2x and then this right here is going to be minus 6 and so this is going to be x squared minus 3 of something plus 2 of something that's minus 1 of that something minus X minus 6 and we've multiplied those two now before we move on and do another problem I want to show you that you can kind of do this in your head as well you don't have to go through all of these steps I just wanted to show you really that this is just the distributive property the fast way of doing it if you had X minus 3 times X plus 2 you literally just want to multiply every term here times each of these terms so you'd say this x times that X so you'd have x squared then you'd have this x times that 2 so plus 2x then you'd have this minus 3 times that X minus 3x and you have the minus 3 or the negative 3 times 2 which is negative 6 and so when you simplify once again you get x squared minus X minus 6 and it takes a little bit of practice to really get used to it now the next thing I want to do and the principle is really the exact same way but I'm going to multiply a binomial times a trinomial times a trinomial which many people find daunting but we're going to see if you just kind of stay calm it's not too bad three X plus two times 9x squared minus six x plus four now you could do it the exact same way that we did the previous video we could literally take this 3x plus two distribute it on to each of these three terms multiply 3x plus two times each of these terms and then you're going to distribute each of those terms into 3x plus two it would take a long time and in reality you'll never do it quite that way but you will get that it the same answer we're going to get when you have when you have larger polynomials the easiest way I can think of to multiply them it's kind of how you multiply long numbers so we will write it like this 9x squared minus 6x plus 4 and we're going to multiply that times 3x plus 2 and what I imagine is when you multiply regular numbers you have your ones place your tens place your hundreds place here you're going to have your constants place your first degree place your second degree place your third degree place if there is one actually there is there will be in this video so you just have to put things in their proper place so let's do that so you start here you multiply almost exactly like you would do traditional multiplication two times four is eight it goes into the ones or the constants place two times negative six two times negative 6 is negative or two times negative 6x is negative 12x so negative 12 X 2 and we'll put a plus there that was a plus 8 two times 9x squared is 18x squared so we'll put that in the x squared place now let's do the 3x part I'll do that in magenta so you see how it's different 3x times 4 3 x times 4 is 12x positive 12x 3x times negative 6x what is that the X is 2 the x times the X is x squared so it's going to go over here going to go over here and then 3 times negative 6 is negative negative 18 and then finally 3x times 9x squared the x times the x squared is X to the third power X to the third power 3 times 9 is 27 I wrote it in the X third place and once again you just want to add the like terms so you get 8 there's no other constant term so it's just 8 negative 12x plus 12x these cancel out 18x squared minus 18x squared cancel out so we're just left over here with 27 X to the third so this is equal to 27 X to the third plus 8 and we are done and you could use this technique to multiply trinomial times a binomial trinomial times the trinomial or really you know you could have five terms up here fifth degree times a fifth degree this will always work as long as you keep things in their proper degree place