Write the first three terms in each binomial expansion, expres...

Question

Write the first three terms in each binomial expansion, expressing the result in simplified form. (x² + 1)¹⁶

Accepted Answer

Hey everyone in this problem we are asked to enumerate the first three terms of the resulting expression. If we were to apply binomial expansion to the following and we have a squared plus two all to the exponent 18. Now let's recall binomial expansion theorem tells us that if we have X plus Y to the exponents end, we can write this as the sum K Equal 0 to N. And juice K X to the n minus K. Why to the exponent? Okay, okay. So this is a way to expand that expression. Let me also. Right. If we can recall the N juice K can be written as N. Factorial over K. Factorial and minus K. Factorial. Ok, alright, So we have this now we need to apply this to our expression. Okay, so let's first start by figuring out what is X in our expression. Okay, so here we have X in our brackets and for us x is going to be a squirt. Similarly, why in our expression is going to be too Okay, we have a plus two. So we have plus Y, Y is two and n. Is that exponent? Which in our case is 18. So we have X, wine and we can go ahead and apply the expansion theorem. Binomial expansion theorem. And use this formula. So we want to find the first three terms. So let's start with the first term in general. The first term. Well, it's going to be given when K is zero. So we're gonna have n choose zero. X. To the exponent N. Why to exponent 0? We found that in our case we have N is 18. So we get 18 choose zero. Okay, X is a square. So we get a squared exponent 18. Why? To the exponent 0 - two. The exponents zero. Now working out 18 to 0 we have 18 factorial divided by zero factorial times 18 minus zero factorial. Ok recall that zero factorial is defined to be one. So here we just get 18 factorial divided by 18 factorial. Ok. So we're left with one there Have a to the and 2 to the exponent zero is just one. So this is going to give us again 18 factor out of 18. Factorial is going to be one. Might just get a to the experiment. 36. So this is our first term. A to the exponent 36. All right now we want to do the second term. Okay, so we started first term when K is zero. Now we're gonna go k up one. Okay, so we have K is equal to one for our second term. So let's go ahead and write out the equation. We have our second term. Pay is equal to one. This is going to give us in general terms and choose one, X to the N -1. Y. to the exponent one. OK in our case and is 18. So we get 18 choose one. Sorry. Okay A which is X? Sorry, a square. Getting ahead of myself N 18 minus 18 times Y. Which is two to the exponent one. Writing out this 18, choose one. We get 18 factorial divided by one factorial times 18 minus one factorial. A square to the exponents 17. And we're gonna have times two. Okay to to the exponent one is just to. All right now We're gonna do a little trick here. So in the denominator we're gonna end up with 17 factorial. One factorial is just 1, 18 -1 factorial. We're gonna have 17 factorial. What we wanna do is we want to get a 17 factorial in the numerator so that we can cancel those terms. Well 18 factorial is just 18 times 17 times 16 times 15. All the way down to one. So we can also write 18 factorial is 18 times 17 factorial Divide by factorial didn't The 17 factorial will divide A. to the exponent 34 times two. And so we have 18 times to a. To the 34 which is going to be equal to 36 a. To the exponent 34. So this is our second term 36 a. To the exponent 34. We have one term left to do. Okay so moving on to the third term, we have K. Is equal to two. So we get the third term and choose to X. to the N -2. Why sweatshirt? Using the values from our problem? We have 18, choose to A squared to the 18 -2 times two squared 18 choose to 18 factorial divided by two factorial times 18 - factorial. A square to the experiment 16 Times two Squared which is four. Alright if we write this out we get 18 using the same kind of trick 18 times 17 times 16 factorial OK because this time we have 16 factorial in the denominator we want to write it to factorial so we can cancel those like terms we get two factorial which is just two times 16 factorial. Those will cancel. We have a two the 32 times four. And if we work this out 18 times 17 times four divided by two. This gives us 612 a. To the exponent 32. So that is going to be our third term. So if we go back up to our answer choices, we see that we have C. Okay. The first term is eight of the 36. The second term is 36. 8 of the 34 the four. Sorry, the third term is 612. 8 to the 32. Thanks everyone for watching. I hope this video helped see you in the next one

Write the first three terms in each binomial expansion, expressing the result in simplified form. (x² + 1)¹⁶

Key Concepts

Binomial Theorem

Binomial Coefficients

Simplifying Powers and Terms

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Write the first three terms in each binomial expansion, expressing the result in simplified form. (x² + 1)¹⁶