In Exercises 9–30, use the Binomial Theorem to expand each binomi...

Question

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (2a + b)^6

Accepted Answer

Hey everyone welcome back in this problem. We are asked to perform binomial expansion and simplify the resulting expression. For the following binomial were given seven M plus N. All to the exponents six. So let's go ahead give yourself some space to work. Now let's recall. Binomial expansion theorem tells us that if we have an expression X plus Y to the exponent N. You can write this as a sum K. from 0 to N. N, choose K. X. To the n minus K. Y. To the K. Okay. Where N choose K as the standard definition and factorial over K factorial times, N minus K. Factorial. Ok. Alright. So let's compare this to our expression and again our expression is seven M plus N. To the exponent six. So in our case X is going to be equal to seven M. Why is going to be equal to N? Okay, and N is the exponent that's going to be equal to six. So substituting in our values we get seven M plus N. To the exponent six is equal to the sum K equals zero 26. I have six choose K. Seven M. To the six -K. And to the K. All right now we want to write this out completely. We're gonna have 6-0. Okay, so K07 m. to the 6 0 And to the zero Plus six Choose 1. Seven M. To the 6 -1 And to the one Plus six shoes too. seven m. to the 6 -2 and squared Plus six Choose 3. I'm seven M to the 6 -3. And cute. Okay and I'm just gonna continue down below adding the rest because we're out of room here We get six choose 4 seven m. to the 6 -4. And to the floor plus six juice. +57 M. To the six minus five. And the five. And the last term n. Is six. We get 6267 M. To the six minus six and to the six. Okay so that is the entire expression right now. Using binomial expansion theorem now we just need to go ahead and simplify. Okay so let's start with all of these um co efficiencies, choose 6 to 06, choose one. Let's let's go ahead and figure out what all of those are. Okay so let's start with 6 to 0. Okay by definition this is gonna be six factorial over zero factorial times six minus zero factorial. Zero factorial is defined to be one. So we just get six factorial over six factorial Which is equal to one. Now for six choose 1 we're gonna get six factorial over one factorial. Six minus 56 minus one factorial. Now in the denominator here we are going to get five factorial. We want to write the numerator to have a five factorial in it so that we can cancel those terms and simplify this and that's a little trick. So if we write six factorial as six times five factorial one factorial is just one and then we have five factorial. Okay so now we can divide the five factorial and we're left with just six. Yeah, Alright six choose to now. Six factorial over two factorial. 6 -2 factorial. Okay, we're gonna do the same thing here but in the denominator we're gonna have four factorial so we're gonna do the same little trick but this time we're going to take it one step further. So we're gonna have six times five times four factorial Divided by two factorial four factorial. We can divide by the four factorial. We're left with Six times five divided by two. Which is going to be 15. Okay. All right, move up a little bit And we're gonna do the rest of the terms. So now we have six choose three And which is going to be six factorial over three factorial times 6 -3 factorial. This is going to be and again we can do the same trick if we want. Um In this case it doesn't make it too much easier but we can do it three factorial Divided by three factorial. Three factorial. So the three factorial will cancel. We're left with six times five times four divided by well what's three factorial three times two times one. That's gonna give us six. And so the six will cancel as well and we're going to be left with 20. Alright for the rest of the terms. You're going to see that we have this symmetry. Okay, so let me go ahead and show you what that's gonna look like when we have six choose four. We had six factorial divided by four factorial times 6 -4 factorial. This is equal to six factorial over four factorial. Two factorial OK, well this is just the same as six choose to. This is the same expression we had were six choose to and so this is just going to be 15. Hm Similarly for six choose 5, We had six factorial over five factorial times 6 -5 factorial OK, so we're gonna have six factorial over five factorial times one factorial well this is just equal to six, choose one. And so you see this symmetry in those terms because we have n and then N minus K. We get this symmetry and you'll see that all of the time when you're doing binomial expansions like this. And finally six choose 6. This is going to be equal to six factorial Divided by six factorial. 6 -6 factorial which is going to be zero. Okay, and this is going to give us one the same as 6-0. Okay, so we have all of our coefficients here. Let's just highlight them so we don't lose track of them. Now we can go ahead and plug these into our expression and simplify. So we get seven M plus N. To the exponent six is equal 6-0 which is one. Okay. seven m. to the 6 0 is seven M. Okay and then end to the zero is just one Plus six choose 1 which is six A. seven M. to the experiment five times and plus six choose to seven M. to the Exponent four. And sward six choose 3 1 7 M. Cubed and Cubed Plus six Juice 4 15 seven M squared. End to the floor. And we'll do the last two terms. We'll add them down below here we're out of room and we get we're on six juice five which is six times seven M. To the exponent one. And to the exponents five plus 6 to 6 which is one seven M. to the zero An end to the six. Okay so one last line to simplify this We're gonna have seven M. Oops we missed the exponent here. We have one. We have seven M. To the exponents six here. So when we do seven to the exponents six we're gonna get 117649. Um to the six plus 100, M. to the five end plus 36,015 K. We're just working out all of these exponents M. To the four and squared. And remember these experiments if you have seven em to the exponent four. That exponent applies to the seven as well as to the M. Okay that's a common mistake Thousands and Nope. I'm cubed and cubed Plus 735 M squared. And to the four plus 42 M. N. to the five plus end to the six. Okay. Alright, so that is the expansion and simplification everybody know me. Let's go back up to the answer choices and we see we have answered. A seven M plus N. To the six is equal to 117,649 M. Two. The six plus 100,842 M. To the five N plus 36,015 M. To the four N squared plus 6860 M. Cube and cube plus 735 M squared. End to the four plus 42 M. N. To the five plus N. To the six. Thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (2a + b)^6

Key Concepts

Binomial Theorem

Binomial Coefficients

Simplification of Expressions

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