Use the Binomial Theorem to expand the binomial and express the r...

Question

Use the Binomial Theorem to expand the binomial and express the result in simplified form. ((x^2)-1)^4

Accepted Answer

Hello today, we're gonna be using the binomial theorem to expand the binomial quantity A squared plus three race to the fourth power. Now, let's just quickly recall what the binomial theorem states. So the binomial theorem states that if you have a binomial quantity in the form of A plus B race to the end power, you can rewrite this as the some nation from K equal to zero to end of the binomial coefficient, N. K times a race. The power of n minus K times B raised to the power of K. And this is also where the binomial coefficient N. K is equal to N. Factorial over N minus K. Factorial times K factorial. And one more thing to note is that zero factorial is equal to one. So let's go ahead and figure out what are A B and n are going to be. So let's take a look at the binomial given to us. So A is going to equal to a squared because a squared is in the slot for A and B is going to be three and n is just gonna be the power that this binomial is raised to. So N is going to equal to four. So, rewriting this in the summation is gonna give me the summation from K equal to 0 to 4 of binomial coefficient 40 times a squared raised to the power of n. Which is four minus K times B, which is three raised to the power of K. Alright, so since we want the simplified form of this binomial, we're gonna need to find the simplified version of all the terms of the summation. So we're gonna want to find all the terms for when K is equal to zero all the way to when K is equal to four. So let's go ahead and start that process. So when K is equal to zero, the binomial expansion is going to be n factorial or four factorial over n minus K or four minus zero factorial times K factorial which is zero factorial times a squared raise the power of four minus K which is four minus zero times three, raised to the power of K which is zero. So simplifying this three raised to the power, zero is just going to be one And 4 0 is just going to be four. So what we have is a squared raised to the power of four which is going to simplify to a raised to the 8th power And 4 0 is just going to be four. So we have four factorial on the denominator and a four factorial on the numerator. So we can go ahead and cancel those quantities and zero factorial is just gonna be one. So what we have is a to the eighth power times one times one which is just going to be a raised to the eighth power and that's gonna be our first term. Now we want to go ahead and repeat this process for K is equal to 123 and four. So when K is equal to one we get four factorial over four minus one factorial times one factorial times a squared raised to the power of four minus K, which is four minus one times three. Race the power of K, which is one now three raised to the power of one is just going to be three and four minus one is going to be three and a squared raised to the power of three is going to be a to the sixth Power now four minus one is going to be three. So we have three factorial and one factorial is just going to be one. So we can go ahead and expand this four factorial to be four times three factorial. And since we have a three factorial on both the numerator and denominator, we can go ahead and cancel those out. So what that leaves us with is four times a race of the power of six times three. And that's going to give me 12 times a race to the power of six. Now, let's go ahead and repeat this process for K is equal to two. So when K is equal to two, we get four factorial over 4 - factorial times two factorial times a squared raised to the power of four minus two times three race, the power of two. So three squared is going to be 94 minus two is just going to be two and a squared raised to the power of two is going to be a to the fourth now four minus two factorial is going to be two factorial and two factorial is the same thing as too. So on the bottom we have two times two which is going to be four and we can go ahead and expand four factorial to be four times three factorial and since we have four on the numerator and the denominator we can cancel those out. So what this leaves us with is three factorial times a race to the power of four times nine. Three factorial simplifies to six and six times nine is going to be 54. So what we have here is 50 for a to the fourth power. Now continuing this for K is equal to three. We get four factorial over four minus three factorial times three factorial times a squared raised to the power of four minus three times three raised to the power of three. Now simplifying this three cubed is going to give me 27. -3 is going to be one and a squared race. The power of one is just a square Now 4 - factorial is just gonna be one factorial which is just one and we can go ahead and rewrite four factorial as four times three factorial. Now we have a three factorial on both the numerator and denominator. So we can go ahead and cancel that out and what that leaves us with is four times a squared times 27 and that's gonna simplify two, times a squared. Now, finally let's go ahead and right out. The expansion for K is equal to four. So when K is equal to four, we get four factorial over four minus four factorial times four factorial times a squared raised to the power of four minus four times three raised to the power of four. So simplifying this three, race to the power of four is going to be 81 And 4 -4 is going to be zero and a squared raised to the power of zero is just going to be one. Now, 4 -4 factorial is just gonna be zero factorial. So this cancels to just one. And since we have a four factorial on both the numerator and denominator, we can cancel both and just get one. So what this leaves us with is one times one times 81, which is just going to be 81 and that's gonna be the final term. So listing everything out, that means that the binomial of a squared plus three, race to the fourth power is equal to a to the eighth plus 12 A to the sixth plus 54 A to the fourth plus 108 A squared plus 81. And that's gonna be the simplified version of this binomial using the binomial theory. So I hope this video helps you in understanding how to use the binomial theory to get a simplified version of a binomial, and I'll go ahead and see you all in the next video.

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