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# Expanding binomials

CCSS.Math:

## Video transcript

so we've got 3y squared plus 6x to the third we're raising this whole thing to the fifth power and we could clearly use a binomial theorem or Pascal's triangle in order to find the expansion of that but what I want to do really is a as an exercise is to try to hone in on just one of the terms and in particular I want to hone in on the term that has some coefficient times X to the sixth Y to the six so in this expansion some term is going to have X - 6 y - 6 and I want to figure out what the coefficient on that term is and I encourage you to pause this video and try to figure it out on your own so I'm assuming you've had a go at it and you might have at first found this to be a little bit confusing I'm only raising it to the fifth power how do I get X to the sixth Y to the sixth but then when you look at the actual terms of the binomial it starts to jump out at you okay I have a y squared term I have an X to the third term so when I raise these two powers I'm going to get I could have power is higher than the fifth power but actually think about which of these terms has the X is 6 Y to the sixth let's just look at the pattern in in I guess the the actual expansion without even thinking about its coefficients and we've seen this multiple times before where you could take your first term in your binomial and you can start it off it's going to start off at a at the power we're taking the whole binomial - and then in each term it's going to have a lower and lower power so let me actually just copy and paste this so this is going to be so copy and so that's first term second term make sure have enough space here second term third term fourth term fourth term fifth term and six terms is going to have six terms to it you always have one more term than the exponent and then actually before I throw the exponents on it let's focus on the second term so the second term actually I'll write like this so the second term is x 6x to the third let me copy and paste that whoops so let me copy and paste that so we're going to put that there that there that there that there and then over three off your screen I have it I wrote it over there and we'll see if we have to go there and then let's put the exponents so this exponent this is going to be the fifth power fourth power third power second power first power and zeroth power and for the blue expression for 6x to the third is going to be the zeroth power first power first power second power third power fourth power and then we're going to have the fifth power right over here then wood and of course they're each going to have coefficients in front of them they're each going to have coefficients in front of them so there's going to be a coefficient in front of this one in front of this one in front of this one then we add them all together but which of these terms is the one that we're talking about it has X to the sixth Y to the sixth so here we have X if we take Y squared to the fourth that's going to be Y to the 8th so that's not it if we go here we have Y squared to the third power that's Y to the sixth and here you have X to the third squared that's X to the six so what we really want to think about is what is the coefficient what is the coefficient in front of this term in front of this term going to be essentially if you put it in this way it's going to be the third term that we actually care about so what is this coefficient going to be now we have to be clear this coefficient whatever we put here that we can use the binomial theorem to figure out is it going to be this this thing that we have to I guess our actual solution to the problem that we posed is going to be the is going to be the product of this coefficient and whatever other coefficients we have over here because we're gonna have three to the third power six squares we're gonna have to figure out what that is but let's first just figure out what this term looks like this term in the expansion what this yellow part actually is and there's a couple of ways we could do that we could use Pascal's try or we could use combinatorics if we use combinatorics we know that the coefficient over here this is going to be 5 choose 0 this is going to be the coefficient the coefficient over here is going to be 5 choose 1 and we've seen this multiple times you could view it as essentially the exponent choose the it's the top the top when we the 5 is the exponent we're raising the whole binomial to and then when we say choose this number that's the exponent on the second term I guess you could say so this would be 5 choose 1 and this one over here the coefficient this thing in yellow this is going to be 5 5 choose 2 5 choose 2 now what is 5 choose 2 well that's equal to 5 factorial over 2 factorial over 2 factorial times times 5 minus 2 factorial so let me just put that in here times 5 minus 2 factorial and this is going to be equal to let's see 5 factorial is 5 times 4 times 3 times 2 we could write x 1 but that won't change the value over 2 factorial actually let me just write that just so we make it clear it is x 1 there 2 factorial is 2 times 1 and then what we have right over here this is 3 factorial so times 3 times 2 times 1 so let's see this 3 could cancel with that 3 that 2 could cancel with that 2 the ones don't matter won't change the value and then 4 divided by 2 is 2 so that is just 2 so we're left with 5 times 2 is equal to 10 so that's going to be this number right over here this is going to be a 10 now another way we could have done it is using Pascal's triangle we could have said ok this is the binomial now this is when I raise it to the second power is 1 to 1 or the coefficients when I raise it to the third power the coefficients are 1 3 three-one when I raise it to the fourth power the coefficients are 1 4 6 4 1 and when I raise it to the fifth power which is what the one we care about the coefficient coefficients are going to be 1 5 10 5 are size 10 10 5 and 1 and we know that when we go for this is going to be the third term so this is going to be the coefficient right over here 1 2 3 third term so either way we know this is 10 but now let's try to answer our original question what is this going to be what are we multiplying times X to the sixth Y to the sixth and now we just have to essentially rewrite this expression so it's going to be 10 times 3 to the third power 3 to the third power times y squared to the third power which is y squared to the third power is y to the sixth power Y to the sixth power times 6 squared so times 6 squared times X to the third squared which that's X to the 3 times 2 or X to the sixth and so this is going to be equal to let's see it's going to be 10 times 27 times 36 times 36 and then we have of course our X to the sixth and Y to the sixth so this is going to be essential you'll see 270 times 36 so let's see we've got a calculator out 270 I could have done it by hand but I'll just do this for the sake of time times 36 is 9,000 720 so that's the coefficient right over here it's going to be 9,000 720 X to the sixth Y to the sixth 9700 20s to the sixth Y to the sixth and we're done