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. (x − 1)^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 were given X -4 to the exponent five. We're just gonna write the binomial. We were given down here X -4 to the exponent five and then scroll down to give ourselves some more room to work. Okay? So when we're giving it by no meal and were asked to expand it, what we want to do is we want to recall binomial expansion theorem. Okay, binomial expansion theorem tells us that if we have a plus B to the exponent end, we can write this as a sum K Equal 0 to N. N, juice K. A to the n minus K. B to the K. N choose K. Has its standard definition. Okay. And tsk music, what's the end factorial over K. Factorial and minus K. Factorial. All right, so we have this general expression that's going to allow us to expand our binomial. Let's go ahead and figure out what A B and n are in our case. Okay, So A is just gonna be the first term in the brackets. In our case it's X. B is the second term. But be careful here, we have plus B. And in our case we have minus four and so B is going to be minus four. Okay, so just watch for the signs there And then N is going to be the experiment, which in our case is five. So we know A. We know B. We know and we have this general um way to expand the binomial. So now we can write it out expanded X minus four. The experiment five is going to be the first term is when K is zero. So we get N. Which is five choose zero. A. Which is X. To the exponent N. Which is five minus K. Which is zero Times B. Which is negative for to the explosion zero. Alright then we move to our next term. K is one. So we have five choose one and we get A. Which is X to the exponent and minus K. Five minus one Times be -4 to the exponent K. Which is one. Okay next we have K. S. 25 choose to X to the 5 -2, negative four squared K. is 3. 5 choose three X. to the 5 -3 negative four. Cute. We're just gonna add the last two terms down here because we're running out of room. We have five choose 4 X. to the 5 -4, negative four to the exponent four. And finally K. S. Five. We have five choose five X. To the five minus five negative four to the exponent five. Alright, so we use binomial expansion theorem to completely expand our expression. Now all we need to do is to simplify. Okay, so we have all of these coefficients 5 to 05, choose 15 choose to. Okay, all of those, we want to Um determine their value. Okay, so that we can simplify this expression. So let's start with each of those. Okay starting with 5-0 and again using the standard formula we have five factorial over zero factorial 5 0 factorial. Now zero factorial is defined to be one. So we just get five factorial over five factorial which gives us a value of one. Five choose one. We have five factorial over one factorial 5 -1 factorial. If you have access to a calculator you're able to use that. That's fine. You can use your calculator for this. I'm going to show you how to approach this if you don't have a calculator and you want to work out these factorial without multiplying a whole bunch of big numbers. Alright so in the denominator here one factorial is just going to be one and then we have four factorial. If we're able to get a four factorial in the numerator then we can divide those out. Well five factorial is just going to be five times four factorial five factorial is five times four times three times two times one. Which is the same as five times four factorial. Now we can divide those four factorial and we're left with just five. Next five choose to Again five factorial divided by two factorial -2 factorial. And we want to do the same thing in the denominator. We're gonna have a three factorial can do the exact same thing. We just have to do an extra step. So now we have five times four times three factorial. They divided by two factorial is just two times one. So that's two and then we have three factorial. Again, the three factorial is divide out. We're left with five times four divided by two. That's 20 divided by two. Which gives us 10. Alright, on the right hand side we're gonna move on to five, choose three. We have five factorial Over three Factorial Times 5 -3 Factorial. And what you'll notice is that we have symmetry. Okay if we look at five choose to we have five factorial divided by two factorial three factorial. And in five choose three we have five factorial divided by three factorial. Two factorial. So these are actually the same, they're equivalent and we get this symmetry because of the K. And then the n minus K in the denominator. Okay, so this is just equivalent to five choose to which we found to be 10. five choose 4. This is gonna be five factorial Over four factorial times 5 -1 factorial 5 - Factorial. Let me rewrite this 5 -4 Factorial. I'm getting ahead of myself. And you'll notice that this looks just like five choose one. Okay we have that symmetry. Again, we have five factorial in the numerator and then we have one factorial times four factorial here and over here we have four factorial times one factorial. So these are the same. Five choose one which we found to be five. In the final five choose five. Five factorial over five factorial times five minus five factorial. This is going to be five factorial divided by five factorial Ok five minus five. We're gonna get zero factorial which is one. And this just gives us one and again we have symmetry. That is the same as 5 to 0. Alright so we found all of those coefficients. We know our equation. Let's try to simplify. So now we have x minus four to the exponent five is equal to 5 to 0 which is one X to the exponent five and -4 to the exponent zero which is just one. five choose 1 was five X to the exponent four and -4 to the one is negative for We get five choose to which is X cubed -4 Squared. That gives us positive 16. And we get five choose 3 which is x squared negative four cubed. Which gives us -64. We got five choose 4. Just five X. to the exponent one Times -4 to the exponent four gives us 256. And you'll notice this alternating positive negative positive negative. Okay we have negative four. So when we raise that to an exponent that is even we get a positive value. If we raise it to an odd exponent we get a negative value. Okay in this final term we're just gonna write down below because we're running out of space we get one X. to the zero and then negative four to the exponent five which is negative. 1024. Okay? So if we write this all out nice and simplified we get X to the exponent five -20 x. to the exponent four Plus x cubed - X squared Plus x -1024. And that is our expanded expression of that binomial. So if we go back up to our answer choices, we see that we have answer B. K x minus four to the experiment. Five can be written as X to the five minus 20 X. To the four plus 160 X cubed minus 640 X squared plus 1280 x minus 1024. 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. (x − 1)^5

Key Concepts

Binomial Theorem

Binomial Coefficients

Simplification of Expressions

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