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. (2x³ − 1)^4

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 five X. The four minus one All to the exponent four. All right now let's just write this binomial down here and then we're going to scroll down and give ourselves some more room to work. Okay? So we'll write it there so that we can refer to it. All right. Now, when we have a binomial and were asked to expand it, we want to recall the binomial expansion theorem. Okay. And what that tells us, is that A plus B to the exponent end? Okay. We have a binomial like this. We can expand it and write it as a sum K Equal 0 to N. Of entries K A to the n minus K and B to the K and N. Juice K Here has its standard definition, which we can write as N. Factorial divided by K. Factorial and minus K. Factorial. All right. So, we have this expression we want to apply it to our problem. So let's figure out what A B and n are. Okay, well, looking at our problem A Well, A is going to be the first thing in the brackets here. In our case it's five X to the four. In the general one we have plus B. In our expression we have minus one. So B is going to be negative one. Ok. Keep track of the sign here. It's important. And it does make a big difference to the final resulting expression. So we want to be sure that we get the right sign here. Okay. And then end is the exponent which in this case is for Okay so we know a. We know B. We know N. And we know this um formula this theorem that we want to use to expand. Let's go ahead and do it five X. To the four minus one. To the exponent four is going to be equal to, we start with K is equal to zero. So we get N 20 which is 4 to 0 K. A. Which is five X. To the exponent four to the exponent four minus zero times B. Which is negative one to the exponent zero. Then we add this is a sum. So we add the next term. The next term is when K is 14, choose one five x. to the exponent four. to the experiment. 4 -1, -1. To the exponent one. Next term K is 24, choose 25 X. To the four. Four minus two minus one squared. Last four, choose 3 five X. to the four. To the exponent four minus three is negative one cubed. And we're just gonna add the final term down below here, we're out of space for choose 45 X. To the exponent four all to the exponent four minus four times negative one to the exponent four. So that's our expanded expression. We've done the expansion now. Now we just need to simplify. Okay we're gonna start the simplification by figuring out what these coefficients 4 to 04 choose 14 choose to um etcetera are Okay, so let's start with 4 to 0. Okay by definition this is going to be four factorial Divided by zero Factorial Times 4 0 Factorial. Okay, zero factorial is defined to be one. So this just gives us four factorial divided by four factorial which is one. We move on and do four choose 1 again. We have four factorial Divided by one Factorial Times 4 -1 Factorial. Okay if you have a calculator and you're allowed to use it, you can go ahead and do that right away here. Okay if not I'm going to show you a little trick on how you can simplify these expressions and in this case it's a four factorial. So to do the full multiplication isn't so bad. Four times three times two times one. It's not so bad. But this trick can really help if you get bigger numbers so you don't have to do all of that multiplication. Alright, so in the denominator we have one factorial which is just one and then we have 4 -1 factorial. So we have three factorial. What we wanna do is we want to rewrite this four factorial in terms of a three factorial. So that we can divide those terms up. We can write four factorial as four times three factorial four times three times two times one. The same thing. Now, the three factorial will divide out or left with four. Alright, moving to four, Choose 2. Okay, for factorial divided by two factorial times 4 -2 factorial. Now this is going to be four times 3 factorial. And in this case we're doing we're going to use a trick a little bit different just because of what we have in the denominator here and the denominator we have two factorial times two factorial, that's just two times two. Okay, we see that. That's four. So we can cancel or divide out that four we get three factorial which is three times two times one. That's gonna give us six. Okay, so again, because the numbers are smaller, they're a little bit easier to manipulate here. Alright, moving on to four, choose three. And what you're gonna see is that we have symmetry in the school officials or choose three is going to be four factorial Divided by three factorial times 4 -3 factorial Well this is four factorial divided by three factorial, one factorial which is equal to four, choose one. Which we found to be four. Okay, so because in the denominator we have K and n minus K. We get this symmetry when those are equal. Okay, Okay, One last coefficient four, choose 4. We got four factorial over four factorial times 4 -4 factorial, this is four factorial over four factorial times zero factorial, this is just gonna give us one. Okay so it's just like 4-0. Okay so now we have are coefficients we have our expression we're just going to plug them in and simplify. So we have five X. To the exponent four minus one. To the exponent four is equal to 4 to 0 which is one five X. to the exponent four. All to the exponent four. Okay and then -1 to the exponent zero gives us one four choose one. We found to be four. We got five x to the four all to the exponent three and then negative one to the exponent one is negative one four Juice 2 is six five X to the four. All squared negative one squared. That's going to be a positive one. Plus four, choose 3 which is four five X. To the exponent four. All to the exponent one negative one cubed. That's going to be negative one odd power Plus four, choose 4. We found to be one five X. To the four to the exponent zero, negative one. To the exponent four is going to be positive one. So that's all of our terms. One last step to simplify. Now be careful when you're applying the exponent here. This exponent of four has to apply to everything in the brackets meaning it has to apply to the five and to the X. To the four. Okay it can't just apply to the X. 24. That's a common mistake. that can happen. So we get 625 X. To the 16 K. X. To the four to the exponent four. You multiply the exponents. We get X. To the 16 minus X. To the exponent 12 X. To the four cubed Plus x. to the experiment. eight and 4 squared minus X. To the four. And that final term is just plus one. And so this is our expanded and simplified expression for our final meal. If we go back to the top we can look at our answer choices and we see that we have answered. D. Okay five X. To the exponent four minus one all. To the exponent four is equal to 625 X. To the exponent 16 minus 500 X. To the exponent 12 plus 150 X. To the exponent eight minus 20 X. To the four plus one. That's it for this one. 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. (2x³ − 1)^4

Key Concepts

Binomial Theorem

Binomial Coefficients

Simplifying Expressions

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