In Exercises 31–38, write the first three terms in each binomial ...

Question

In Exercises 31–38, write the first three terms in each binomial expansion, expressing the result in simplified form. (x+2)^8

Accepted Answer

Hey everyone! Welcome back in this problem, we are asked to enumerate the first three terms of the resulting expression If we were to apply binomial expansion to the following were given X plus 13. All to the exponent six. Now let's recall binomial expansion theorem tells us we have an expression A plus B to the exponents end. We can write this as a sum K equals 0 to end of N juice, K A. To the experiment and minus K. B to the experiment K N juice K is equal to N. Factorial over K. Factorial times N minus K. Factorial by definition. Okay. All right. So we have this general form for binomial expansion. Let's go ahead and figure out what A B and N will be in our expression. So A in this case is the first thing in our brackets which is going to be equal to X. B. Well, B is the second term in our brackets. In this case we have 13, So B is going to be 13 and N is going to be the exponent. So in this case N is equal to six. Okay, So we have the general form for expansion, we know what A B and n. R. Let's go ahead and find our first three terms. The first term. Well, we start when K zero. So the first term is going to be when K is zero and we're gonna have and 20 A to the experiment and zero B to the experiment zero. This is going to be in our case 6- X. to the exponent 6 0 and 13 to the experiment zero 620 is going to be six factorial divided by zero factorial times six minus zero factorial zero factorial is defined to be one. So here we have six factorial divided by six factorial. This whole term is just going to give us one. We have X to the five and 13 to the exponent zero. Anything to the exponent zero is going to be one. We just have times one and this and sorry not X to the five X to the 66 minus zero is six. Okay so we have one times X to the six times one. This gives us X to the exponent six for our first term. Moving on to our second term For our second term. We're gonna have cables one. Okay we're increasing K by one each time. So we get and choose one A. to the exponent and -1 b. to the exponent one. Let's give ourselves some more room to work and filling in the values for our problem we have N. Is six. So we get six, choose one X. to the 6 -1 Times 13. 2 Exponents one. This gives us six factorial Divided by one factorial times 6 -1 factorial Next to the Exponent five times 13. Mhm. Now here we're gonna get a five factorial in the denominator. Okay 6 -1 is going to get five. So we have five factorial let's go ahead and try to write five factorial in the numerator as well. So that those terms divide We can write six factorial as six times 5 factorial. Okay. And then we're dividing by one factorial which is just one times five factorial. Okay now the five factorial will divide out and we're left with just six. So we have six times X. To the five times 13 which gives us a second term of 78 X. To the exponent five. Alright, we have two terms done. We have one left to find give ourselves some more room and let's go ahead and find the third term. Now the third term is going to occur one K. Is equal to two. Okay so we have and choose to A to the exponent and -2 B squared. Using our numbers we get six, choose to X. to the exponent 6 -2 times 13 squared six, choose two is going to be six factorial divided by two factorial times six minus two factorial X. to the four and 13 square which gives us 169. Okay, similarly to what we did for the second term and then denominator here we're gonna have four factorial. We want to read the numerator so that we also have a four factorial term so that we can cancel those. Okay this is going to give us six times five times four factorial Divided by two factorial is two times four factorial We have X to the four times 169. If you have your calculator on your test your if you're allowed to use your calculator you can go ahead just plug that in right away. Use your calculator. But this is a good trick to know how to simplify these. Factorial if you don't have access to a calculator. Okay? Alright so the four factorial will divide. We're left with six times five divided by two. Okay so 30 divided by two which is 15 times 169. We get 2535 X. To the fore. And that is our third term. So going back up to the answer choices, we see that we have answer. A The first term is X. To the exponent six. The second term is 78 X. To the exponent five. And the third term is 2535 X. To the exponent four. Thanks everyone for watching. I hope this video helped see you in the next one

In Exercises 31–38, write the first three terms in each binomial expansion, expressing the result in simplified form. (x+2)^8

Key Concepts

Binomial Theorem

Binomial Coefficients

Simplification of Expressions

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