In Exercises 31–38, write the first three terms in each binomial ...

Question

In Exercises 31–38, write the first three terms in each binomial expansion, expressing the result in simplified form. (y³ − 1)^20

Accepted Answer

hey everyone in this problem we are asked to enumerate the first three terms of the resulting expression for to apply binomial theorem and the expression were given is B squared minus two all to the exponent 20. So we're told to use binomial expansion. Let's recall what binomial expansion theorem tells us. If we have X plus Y to the exponent end. Binomial expansion tells us that we can write this as the sum K From 0 to N. Of n choose K X to the exponent and minus K. Y to the exponent K. And this N choose K. This combination here is equal to end factorial over K. Factorial and minus K factorial. Okay, alright. So this is a binomial expansion. This is gonna tell us how to expand the expression and we want the first three terms. So let's start with the first term. Well the first term is going to be when K is zero. Okay, when k is zero we're gonna have n choose zero. X. To the end. Why? To the 0? Okay, that's our first term. Now in this case R X. It's going to be b squared. Okay. Ry is going to be negative too. Okay, be careful of the sign here we have b squared minus two. So are Y. Is negative two. And our end is that exponent? Which in this case is 20. So plugging in these values are first term is going to be 22:00 X. Which is b squared to exponent n. 20 And why? Which is -2 to the exponent zero. Well, 22 0. This is going to be 20 factorial divided by K which is zero factorial 20 minus zero factorial ok recall that zero factorial is defined to be one. Okay, so this is just going to be 20 factorial over 20 factorial. This is gonna give us one B to the exponent and times one. Okay. Anything to the exponent zero is one. And so this gives us a first term of B to the exponent 40. Alright great. So we've done the first term, let's move on to the second term. Now the second term Is going to be when K is one. Okay. In general we're gonna have the second term is and choose one X. to the N -1. Wide to the exponent one. Okay. In our case recalled. And is 20 Okay X. Is B squirt? And why is -2? Well this is going to be 20 factorial Divided by one factorial times 20-1 pictorial, B squared to the 20-1. Well that's gonna be b squared to the times negative two. Now we're gonna do a little trick with the numerator here. 20 factorial recall is 20 times 19 times 18 times 17. All the way down to one. And so 20 factorial we can write as 20 Times 19 factorial OK. In the denominator we have 19 factorial which is why we want to write the numerator in this way. So that we can divide these 19 factorial. Now we have B to the 38 times negative two. Okay, multiplying these together, we're gonna have negative 40 Maybe to the 38 and that is going to be our second term. Okay, so two terms down, one to go moving on to our third term in this case K is too Okay. So in general we get n choose to X to the n minus two, Y squared. Using the values from our problem, we have 20 choose to X. Which was b squared to the exponent 20 minus two times Y negative two squared choose to. It's going to be 20 factorial. We were two factorial times 20-2 factorial. Then we get B squared to the exponents 18 and then minus two squared is going to give us positive four. Okay, we're going to do the same trick this time. So in the denominator we're gonna get 18 factorial. So we're gonna do the same trick, we're just gonna do it twice. So we're gonna have 20 times 19 times factorial in the numerator. And then two factorial is just going to be two And 18 factorial. Again we can divide by the 18 factorial. We have B to the 36 times four. And when we work out these coefficients multiply them together, divide by two. We're going to get 760 B to the exponents 36. And that is going to be our third term. So if we go back up to the top and look at our answer choices, we're going to see that we have answered D. The first term is B to the exponent 40. The second term is negative 40 B to the exponent 38 the third term is 760 B. 2 36. Thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 31–38, write the first three terms in each binomial expansion, expressing the result in simplified form. (y³ − 1)^20

Key Concepts

Binomial Theorem

Binomial Coefficients

Simplification of Expressions

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