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 going to be using the binomial theorem to expand the binomial quantity. So before we jump into this, let's just quickly recall what the binomial theorem is. And it states that if you have a binomial quantity A plus B race of the power of N. This could be rewritten as the summation from K equal to zero to end of your binomial constant. And comic A times a race the power of n minus K times B raised to the power of K. And this is where the binomial constant and kama Kay is equal to N. Factorial over n minus K. Factorial times K. Factorial. And one thing to note as well is that zero factorial is equal to one. So we're gonna go ahead and use this to figure out what the first four terms of this expansion is going to be. So let's just take a look at what's given to us. The binomial quantity is B plus three. Race to the seventh power. So be takes the place of our a variable. So we're gonna let a equal to B. And three takes the place of RB variable. So we're gonna let B equal to three. And finally n is going to be the exponents that the binomial quantity is raised to and that's going to be seven. So N is equal to seven. Now we're gonna go ahead and rewrite this as a summation. So the summation is going to be K equal to zero because we always let this start at zero all the way to end which is seven. And we're gonna have our binomial constant 70 times A. Which is B raised to the power of N. Which is seven minus k times B which is three raised to the power of K. Alright, so in order to write out the first four terms, we're gonna want to write out the terms where K. Is equal to zero and work all the way to where K. Is equal to three. Because this series always starts at zero and zero is considered a term. So let's go ahead and start that process. So when K is equal to zero we're gonna first expand our binomial constant and that expansion is going to be seven factorial because N factorial over N minus K which is seven minus zero factorial times K. Factorial which is zero factorial. And that's going to be multiplied by B raised to the power of seven minus K. Which is seven minus zero times three raised to the power of K. Which is zero. Alright, so that's a lot to take in. But we're just gonna go ahead and simplify little by little. So three race, the power of zero is just gonna be one And be race, the power of 7 0 is going to be be raised to the power of seven because 7 zero is 7 And 0 factorial is just gonna be seven factorial. That means that we're going to have a seven factorial on the numerator and on the denominator. So we can go ahead and cancel these quantities right away and once again zero factorial is just one. So this is gonna end up being 1/1 times B raised to the seventh times one which is going to be B to the seventh. And that's gonna be our first term for this expansion. Now let's go ahead and continue this when K is equal to one, so when K is equal to one, we get seven factorial because N factorial over n minus K which is seven minus one factorial times K factorial which is one factorial and all of that is being multiplied by B. Race to the power of seven minus K which is seven minus one times three. Race of the power of K which is one. So be race. The power of seven minus one is the same thing as be race. The six power three race, the power of one is just going to be three and 7 -1 factorial is going to be six factorial And one factorial is just gonna be one. So rewriting this is going to give us seven factorial over six factorial times one which is six factorial times B raised to the sixth. Power times three which is three B to the sixth. And we can go ahead and rewrite or expand seven factorial To be seven times 6 factorial. And doing so allows us to cancel out a six factorial from the numerator and denominator. So what we're left with is seven times three times b to the six power. And that's going to give me 21 times b to the sixth power. Which is going to be our second term. Now continuing this for K is equal to two. We get seven factorial over seven minus two factorial times two factorial times B raised to the power of seven minus two Times three. Race to the power of two. So three squared is just gonna be nine And be race the power of 7 -2 is gonna be b to the 5th power and 7 -2 factorial is going to be five factorial And two factorial is going to be two times 1. Which is just to So rewriting everything is going to give me seven factorial over five factorial times two times B to the fifth times nine which is nine B to the fifth power. And we can go ahead and expand seven factorial to be seven times six times five factorial. So rewriting seven factorial is going to give me seven times six times five factorial. And doing so allows me to cancel out the five factorial from both the numerator and denominator. So what we're left with is seven times six times nine B to the fifth power. All of that divided by two. And that's gonna give me 189 b to the 5th power. Now we're gonna do this one more time for our 1st 4th term which is going to be K. is equal to three And for K is equal to three. We get seven factorial over 7 - factorial times three factorial times B raised to the power of seven minus three times three. Cute. Okay so simplifying three cubed Is going to give us Be raised to the power of 7 -3 is going to give me B to the 4th and 7 -3 factorial. It's gonna give me four factorial and we have three factorial as well. So simplifying this is going to give me seven factorial over four factorial times three factorial times 27 times B to the fourth which is going to be 27 B to the fourth power. And we can go ahead and expand seven factorial to be seven times six times five times four factorial. So quickly rewriting this is going to give me seven times six times five times four factorial. And doing so allows me to cancel out the four factorial from both the numerator and denominator and three factorial is the same thing as six. So what we are left with is seven times six times five times 27 B to the fourth. All of that divided by six. And that's gonna give me 945 B to the fourth. So let's just go ahead and list out all of our terms. So the first term we came up with is B to the 7th. Then we got 21 b to the 6th, than 100 89 B to the fifth. Then finally 945 B to the fourth. And that's going to be the first four terms of the binomial expansion, which also means that the answer is going to be D. So I hope this video helps you in understanding how to do binomial expansion and I'll go ahead and see you all in the next video.

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