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. (x - 2y)¹⁰

Accepted Answer

Hey everyone! Welcome back 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 church minus three W. All to the exponent. 12. Now let's recall binomial expansion tells us that if we have an expression X plus y to the experiment end, we can expand this and write it as a sum Kate. Will 0 to end. N choose K. X. to the N -K, Y two K. And also recall the N choose K. We can write this as N factorial over K. Factorial and minus K. Factorial. Ok, Alright. So that's our general form. Now let's figure out what is X. Y. And and in our case. Okay, well in our case X is just going to be a church. Okay. This first term in our brackets. Okay, why? Why is the second term? And in this case it's negative three W. Okay. So why is going to be negative three W And R. N is going to be the exponent, which in this case is 12. Okay, so we know our X. We know why we know our end. We have our general formula. Let's go ahead and find the first three terms that were asked to find. Okay, so the first term this is gonna occur when K is equal to zero. And so we have n choose zero. X. to the N zero. Why to the zero Writing this in terms of our numbers from our expression we get 12-0 Each to the exponent. 12 minus zero And -3 W. to the exponent zero. All right well 12-0 is going to be 12 factorial divided by zero factorial? 12 0 factorial. OK so we have 12 factorial divided by 12 factorial. That's just gonna be one. H. to the 12 times one. Okay we have negative three W. To the exponent zero. Anything to the exponent zero is just going to be one. And so this is going to leave us with just a church to the exponent 12. So that's gonna be the first term in our expression. When we expand. Moving on to the second term, the second term we're gonna have K equals one. Okay moving up one. So in general we get and choose one X. to the N -1 Y. to the exponent one. Giving yourself some more room to work here. Alright, substituting in the numbers we have. We get that. This is going to equal 12, choose one. H. to the exponent 12 -1 -3 W. to the exponent one This is going to give us 12 factorial divided by one factorial. 12 -1 factorial H. to the exponent. 11 times negative three W. Alright in the denominator we're gonna get an 11 factorial. So let's go ahead and see if we can write 11 factorial in the numerator in order to cancel those terms. And simplify this expression. Well we can write 12 factorial as 12 times 11 factorial And then we have one times 11 factorial in the denominator. And those 11 factorial will divide and we're just left with 12. We have a church to the 11. That was -3 W. So we end up with negative three times 12 which is gonna give us negative 36 H. two. The explosion of 11 times w. And so that is going to be the second term in their expression when expanded. We've done two terms, we have one left to do. So our third term Is going to occur when K. is equal to two. And in general we're gonna have end choose to X. to the N -2. Why do the exponents? Okay. Using the values from our problem we get 12 choose to H. to the 12 -2 times negative three W squared. Okay. All right. And again 12 years two. We have 12 factorial divided by two factorial times 12 -2 factorial. HD explanation. 10. Now negative three W squared. We have negative three squared. Which is going to give us nine and then R. W squared. Now we're gonna use kind of the same trick that we used above. But in this case we have 10 factorial in the denominator. So instead of stopping when we have the 11 factorial we're gonna go one step further. So we're gonna write 12 factorial as 12 times 11 times 10 factorial. All Divided by two Factorial Times 10 Factorial. Okay the 10 factorial will divide out. We're left with 12 times 11 divided by two H. Two the 10 times nine W squared. And we do 12 times 11 times nine divided by two. This is going to give us H. To the 10 W squared. Okay. And so that is our third term. So now we found the first three terms of our expansion. Let's go back up to our answer choices and we'll see that we have answered. Be the first term is H. To the exponent 12. The second term is negative 36 H. To the 11 W. And the third term is 594 H. To the 10 W. Squared. Thanks everyone for watching. I hope this video helped see you in the next one.

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Write the first three terms in each binomial expansion, expressing the result in simplified form. (x - 2y)¹⁰