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. (c+2)^5

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 C plus four to the exponent five. So we're gonna write that down below C plus four. The exponent five. Give ourselves some room to work. Now when we're given a binomial and we want to expand it. We want to recall the binomial expansion theorem. Okay. And that tells us that if we have a binomial, X plus Y to the exponent and we can expand it and write it as a sum K equals zero to end of N, choose K. X. To the n minus K. Y. To the K. K. Where n choose K has its standard definition of N factorial divided by K. Factorial, N minus K. Factorial. Ok, Alright. So we have this expression that we can use for our binomial. Let's figure out what X, Y and N is in our case. Okay, so X is going to be the first thing in this C plus four terms, it's gonna be C. Why we have plus y here we have plus four. So y is going to be four and n is the exponent which in this case is five. So we know the expression we need to use to expand, we know what X. Y and n. R. Let's go ahead and apply it to our binomial C plus four to the exponent five is going to be, we start with K equals zero. We have N which is five, choose K which is zero X. Which is C to the exponent 5 0 times y. Which is 4 to the exponent zero. We have a some. Okay so we add the next term. The next term is one K. Is 15 choose one C. To the exponent five minus 14 to the exponent one Add the next term five choose to see to the experiment. 5 -2. Four squared. Next term five choose three C. To the five minus 34. Cute. Hey I'm gonna add the last two terms down here just because we're running out of space we get five choose four C to the Exponent 5 -4. 4 to the exponent for Plus five choose 5 C. to the exponent 5 -5. 4 to the exponent five. Okay. Alright so we've expanded our expression. Now what we have left to do is to simplify. So let's start with these coefficients. Okay we have 5 to 05 choose 15 choose to all the way to five choose five. Let's figure out what those are. And then that will allow us to simplify this expression. Alright so we have 5-0. And going from definition the definition we wrote at the top right? This is gonna be five factorial divided by zero factorial times five minus zero factorial OK and zero factorial is defined to be one. So this is just five factorial divided by five factorial. This gives us a value of one. All right now the 2nd 15 choose one. This gives us five factorial divided by one factorial times five minus one factorial OK, alright. And if you have a calculator, if you're able to use a calculator, you can go ahead and use a calculator right off the bat from here. Okay, But I'm gonna show you a little trick on how to deal with these things if you don't have access to a calculator. Okay. All right, so one factorial is just going to be one and then we have 5 -1 factorial which gives us four factorial in the denominator. Okay, so if we can write the numerator to have a four factorial term that will divide out. And it will make it simpler to do this calculation. Well we can write five factorial as five times four factorial, Five factorial is five times four times three times two times 14 factorial is four times three times two times one. Okay, so five factorial can be written as five times four factorial. Four factorial will divide. We're left with just five. Alright, next term five choose to Five factorial divided by two factorial, 5 -2 factorial. And we're gonna do the same thing on the bottom we get two factorial which is two times one. This just gives us two. We have three factorial and then the numerator five factorial we're gonna again do the same thing but one step further times four times three factorial. Ok. The three factorial will divide. We're left with five times four divided by two. That's 20 divided by two. That gives us 10. And in this case when we're dealing with five factorial, the numbers aren't that big to multiply together. Um So you could just multiply them and divide if you if you wanted to but this is a really great trick if you get into bigger numbers um to kind of manipulate them so you don't have to do multiplication with big numbers division with big numbers. Okay, that can get a bit messy. Alright, next 15 choose three. This is going to be five factorial over three factorial times five minus three factorial. And what you'll notice now is that we have symmetry in these coefficients. Okay, this is five factorial divided by three factorial times two factorial, That's the same as five choose to. Okay, because we have K and n minus K in the denominator we get this kind of symmetry when those things are equal so we have five. This is just equal to sorry five choose three equal to five choose to which we just found to be 10. Okay, five choose four. five factorial over four factorial, 5 -4 factorial and this is going to be five factorial divided by four factorial times one factorial This is the same as five choose one Which is equal to five. And the last coefficient five choose 55 factorial divided by five factorial five minus five factorial which gives us zero. Okay well this is equal to 5 to 0. It is equal to one. So we have the value of all six of those coefficients. We have our expression let's go ahead and plug them in and simplify. We're gonna get C plus four to the exponent five is equal to 5 to 0 which is one C. To the exponent five and 4 to the zero which is one good. Next we get five choose one which we found to be five. We get C to the exponent four times four to the exponent one which is four. Next we get five choose to which we found to be 10. Get C to the experiment three but yet four squared which is 16. Next Term five Choose 3 We found to be 10. We have C squared Times 4 to the exponent three. Just 64. Be at five choose four which is five C. To the experiment one And 4 to the Exponent four which is 256. Hey we're just gonna add the last term down here, we're out of space. Five choose five is one C. To the exponent zero. Gives us one and 4 to the experiment five gives us 1024. Okay alright so we're almost there we've put in those coefficients now we just need to simplify one last time And we're gonna find we get C to the exponent five Plus 20 c. to the exponent four Plus c. cute plus 640 C squared. That's 1280 c Plus 1024. And that is the expanded and simplified version of our binomial that we were given. So let's go back to the top. Take a look at our answer choices. What we see is that we have answers. C. A C plus four to the exponent five is equal to C. To the five plus 20 C. To the four plus 160 C. Cubed plus 640 C squared plus 1280 C plus 1000 and 24. Thanks everyone for watching. See you in the next video.

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (c+2)^5

Key Concepts

Binomial Theorem

Binomial Coefficients

Simplification of Expressions

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