Write the first three terms in the binomial expansion, expressing...

Question

Write the first three terms in the binomial expansion, expressing the result in simplified form. (x-3)^9

Accepted Answer

Hello. Today, we're gonna be using the binomial theorem to find the first four terms of the given binomial. Now, before we jump into this, let's just quickly recall what the binomial theorem states. And that states that if you have a binomial quantity A plus B race of the power of N, this can be represented as the sum nation from K is equal to zero to end of the binomial coefficient, N K times A to the power of n minus K times B to the power of K. And this is also where the binomial coefficient and K can be represented as N factorial over n minus K. Factorial times K factorial. And one thing to note about this, zero factorial is equal to one. Now let's just go ahead and take a look at the given binomial. The given binomial is the quantity B plus three raised to the power of seven. And using our binomial theorem, we can allow a two equal to B. B to equal to three, An end to equal to seven. And plugging this in to our binomial theorem is going to give us the sum nation From K is equal 0-7 of the binomial coefficient seven K times A. Which is B race of the power of N, which is seven minus K times B, which is three race of the power of K. And we're going to want to find the first four terms of this summation. And in order to do that, we want to evaluate this from when K is equal to zero. All the way to when K is equal to three because K is equal to zero is considered the first term. So let's go ahead and expand this when K is equal to zero. When K is equal to zero. We first start by expanding our binomial coefficient, which is going to be n factorial, which is seven factorial over n minus K, which is seven minus zero and seven minus zero is just going to give us seven that quantity factorial. So seven factorial and that's going to be times K factorial and in this case here, K zero. So we have zero factorial. Now keep in mind zero factorial is equal to one. So we can go ahead and replace zero with one and this is going to be multiplied by B raised to the power of seven minus K, which is going to be seven minus zero. So what we are left with is seven minus zero, which is seven. So B to the power of seven times three raised to the power of K, which in this case is zero and anything raised to the zero power is always going to be one. So we have B to the power of seven times one. Now we just need to go ahead and simplify while seven factorial times one is just going to give us seven factorial. So we can go ahead and rewrite this denominator as just seven factorial and notice that we have a seven factorial in both the numerator and denominator, this can just cancel out leaving us with one and one times B to the seventh times one is just going to give us B to the power of seven and this is going to be the first term of our summation. Now we want to go ahead and repeat this process for K is equal to one when K is equal to one, we first again expand our binomial coefficient which is going to be seven factorial over n minus K which is going to be seven minus one this time because K is equal to one, so seven minus one factorial times K factorial which is just one factorial and that's gonna be multiplied by B to the seventh. Power minus k which is B to the seventh minus one, Times 3 to the power of K which is 3 to the power of one. Now we need to go ahead and simplify this one. Factorial is just going to give us one. So we just have one and seven minus one is going to give us six. So we have six factorial times one which is just going to be six factorial, 3 to the power of one is just three and b to the seventh minus one. Power seven minus one is going to give us six. So we have B to the power of six. So simplifying this, we have seven factorial over six factorial times B to the six times three. Now there is a way to rewrite seven factorial, recall that when you have n factorial, this is essentially a product of a decreasing series. So N factorial is represented as n times n minus one, times n minus two times n minus three and so on with each value decreasing by one. So we can rewrite seven factorial as seven times six factorial and notice that we have six factorial in both the numerator and denominator. So we can go ahead and cancel those out. And that leaves us with seven times three times B to the power of six. And that's going to give us 21 B to the sixth and that's going to be the second term for our binomial X. For the first four terms of the binomial theorem. Now we're going to repeat this process again for K is equal to two when K is equal to two, we get seven factorial over n minus k, which is going to be seven minus two factorial times K factorial, which is two factorial in this case times B raised to the power of n minus k, which is going to be seven minus two times three. Raised to the power of K which is two now three race of the power of two is three times three, which gives us nine and be raised to the power of seven minus two, while seven minus two is just going to give us five. So we can rewrite this as B to the power of five Now two factorial is just two times one which is just going to give us two And we have 7 -2 factorial Which is going to be five factorial And we still have the seven factory in the numerator. So rewriting this whole thing is going to give us seven factorial over five factorial times two times B to the power of five times nine. Now, once again we can go ahead and expand the seven factorial. So that way we can get rid of the five factorial in the denominator and we can expand this by writing it as seven times six times five factorial. Now notice that we have a five factorial in both the numerator and denominator. That's just going to cancel out and we are left with seven times 6/2 times nine times B to the power of 57 times six times nine times divided by two is going to give us 189 and 189 times B to the fifth is going to give us B to the fifth and this is going to be the third term Now. Finally we're on the fourth term which is going to be when K is equal to three. So when K is equal to three, we get seven factorial over n minus K, which is seven minus three factorial times K factorial which is going to be three factorial times B raised to the power of n minus K. Which in this case is going to be seven minus three times three. Race of the power of K which is three. Now three to the power of three is three times itself three times so three times three gives us nine and nine times three gives us 27. So three to the power of three, it can be rewritten as 27 B. Race the power of seven minus three while seven minus three is going to give us four. So this can be written as B to the power of four And same thing here. 7 -3 factorial is going to give us four factorial and what we have here is seven factorial over four factorial times three factorial while three factorial can be evaluated as three times two times one. So that's going to give us six and so rewriting and simplifying everything here is going to leave us with seven factorial over four factorial times six times B to the power of four times 27. Now, once again, let's go ahead and expand seven factorial so that we can be able to get rid of the four factorial in the denominator we can expand seven factorial to be seven times six times five times four factorial. And since we have a four factorial in both the numerator and denominator. This is just going to cancel out furthermore, we have a six in both the numerator and denominator. So we can cancel out the six. So what we are left with is seven times five times 27 times b to the power of four. And that is going to give us 945 b to the power of four, which is going to be our fourth term. Now, let's go ahead and combine all the terms together. The first four terms of our original summation is going to be B to the 7th, 21 B to the 6th. 189 B to the 5th and 945 B to the fourth. So this is the first four terms of our binomial using the binomial theorem. And that means that the answer to this problem is going to be D. So I hope this video helps you in understanding how to use the binomial theorem. And I'll go ahead and see you all in the next video

Write the first three terms in the binomial expansion, expressing the result in simplified form. (x-3)^9

Key Concepts

Binomial Theorem

Binomial Coefficients

Simplification of Terms

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