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. (y-3)^4

Accepted Answer

Hey everyone in this problem we are asked to perform binomial expansion and simplify the resulting expression for the following. Binomial were given a -9 to the exponent four. All right, let's recall our binomial expansion theorem. Binomial expansion theorem tells us that if we have X plus y to the exponent and we can write this as a sum K equals 0 to end of n juice, K X to the end minus K. Y. To the K. N. Juice. K is the standard definition and factorial over K. Factorial and minus K. Factorial. Ok, alright. So this is a general equation. Now we want to write the term the binomial that were given in terms of these X, Y and N. Okay, While we're given this here a -9 to the exponent four. Okay, so in our case X is going to be equal to a. Okay, the first term in those brackets, why is going to be equal to negative nine? Okay, here we have plus Y. In our case we have minus nine. So why is negative nine and n is the exponent? Which in our case is four. So now we know our X. Y and N. We can go ahead plug these into our equation. Um for binomial expansion theorem to figure out whatever binomial will be a minus nine to the exponent four will be equal to first term. We're gonna have K is zero. So we're gonna N 4 to 0 X. Which is a to the exponents. End four minus K zero Times Y, -9 to the experiment. zero. Now we have K. Is one. So we have four choose one A. to the exponent for -1 -9 to the Exponent one. K. is 2. 4 choose two A. To the exponent for minus two times minus nine squared Plus four. to choose three A. to the exponent 4 -3 -9 to the exponent three. Okay in the last term K is 4. 4 choose four A. to the 4 -4 times -9 to the exponent four. Okay alright so that is the expansion now what we wanna do is go ahead and simplify this. Okay so let's start with all of these coefficients K. 4 to 0 fortunes. One fortress to 4 to 34 to four. Let's work those out separately and then go ahead and put them back into our equation. Okay so 4 to 0 recall that standard definition And factorial over K. Factorial and minus K factorial. So this is going to be four factorial Over zero Factorial Times 4 0 Factorial. Okay well this is just four factorial over four factorial OK zero factorial recall is defined to be one. So we get four factor over four factorial we just get one. Hm Now four choose 1. Similarly. four factorial over one factorial or - factorial. Okay in the denominator we're gonna have a three factorial. Okay so let's use this little trick. We want to write a three factorial in the numerator as well. So that we can divide those out so we can write four factorial. Well this is four times three times two times one. So this is four times three factorial divided by one. Factorial is just going to be one. And then we get three factorial. And so now we can divide that three factorial. We're left with four. Alright, four Choose 2. Four factorial Over two Factorial Times 4 -2 Factorial. Well this is going to be four factorial over two. Factorial is just to Okay and here we're gonna have two factorial again which is just going to be too. So we have four factorial over four. This is going to give us six. All right onto four choose 3. In this case we get four factorial over three Factorial Times 4 -3 Factorial. This gives us four factorial over three factorial times one. And you'll notice that this is the exact same as four, choose one. Okay, and when we do by normal expansion we have this symmetry among these coefficients. Okay, so this is going to be equal to four, choose one which is just equal to four but because we have that N or the K and then the n minus K. And we get symmetry when they're reversed. Alright, one more coefficient four, choose four. And again you'll see the symmetry here we have four factorial over four factorial times four minus four factorial Ok so this is gonna look just like 4 to 0. We have four factorial over four factorial times zero factorial Ok this looks just like 4 to 0 and that's going to be equal to one. Okay so these are all of our coefficients that we're gonna use. We're gonna plug them in to simplify our equation or expression. So now we have a -9 to the exponent four is equal 4-0. Is one. A. To the exponent for zero is a to the exponent four and -9 to the exponent zero. That's just one. Therefore choose one is 4 A to the 4 -1 is a cute. Okay and then we have times negative nine to the exponent one. Times negative nine. Alright. four choose 2. We have six A. to the 4 -2 is going to be a squared and then we have negative nine squared. Okay, when we have a or an even exponent we're going to get a positive. So negative nine squared is going to give us positive or choose three is going to be four a. to the 4 -3. That's just a Okay and negative nine cubed. It's going to give us -729. Okay. Again odd exponents. So we get a negative and then we have four choose four which is 18 to the four minus four. That's a to the zero. Just going to give us one. We have negative nine to the exponent four which is going to be 6561. Okay. Alright so one more line just to fully simplify this. We get 824. Okay. Four times negative nine is gonna give us negative 36. So we get negative 36. A cute Six times 81 is going to give us 486. We have a squared four times -2916. And we have a and finally this last Constant 6561. Okay. And so this is our simplified expression and if we go back up to our answer choices we will see that we have answered A a a a a minus nine to the experiment four is equal to aid to the exponent four minus 36. 8 cubed plus 486 A squared minus 2916 A plus 6561. Thanks everyone for watching. I hope this video helped you in the next one

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (y-3)^4

Key Concepts

Binomial Theorem

Binomial Coefficients

Simplification of Expressions

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