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 write the binomial in simplified form. So before we jump into this, let's just quickly recall what the binomial theorem states. What it states is if you have a binomial in the form of A plus B race to the power of N, this can be represented as the summation from K is equal to zero to N of the binomial coefficient and 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 keep in mind is that zero factorial is equal to one. So let's just go ahead and take a look at the given binomial. The given binomial is a squared plus three race, the power of four. So here using the binomial theorem, we can let a equal to A squared, we're going to let B equal to three and we're going to let N equal to four. So using this, we can represent this as the summation from K is equal to 0 to 4 of the binomial coefficient four K times A which is A squared raise the power of N minus K, which is going to be four minus K times B to the power of K which is going to be three to the power of K. Now, in order to write this in simplified form, we need to find all the terms of this summation starting from K is equal to zero, all the way to K is equal to four. So we're going to be expanding the summation from when K is equal to zero, going all the way to when K is equal to four. So let's go ahead and start that. We're gonna start when K is equal to zero. And we start this by first expanding our binomial coefficient. The binomial coefficient is N factorial which in this case is going to be four factorial over N minus K which is going to be four minus zero because K is equal to zero, that quantity factorial times zero factorial and that's gonna be times A squared to the power of four minus K. So A squared to the power of four minus K, which in this case is zero times three race to the power of K which is zero. And now we need to go ahead and simplify any number to the power of zero is always going to be one. So three to the power of zero can be represented as one A squared to the power of four minus zero. While four minus zero is just going to give us four. And using our exponential multiplication A squared to the power of four is going to give us a to the power of eight. And on the left hand side here, zero factorial can be represented as one. So zero factorial is just one and four minus zero. Factorial is just going to give us four factorial. But notice that we have a four factorial in both the numerator and denominator. So we can go ahead and cancel out the four factorials making this entire quantity one. And what we are left with is one times eight to the power of eight times one. And that's just going to give us a to the power of A which is going to be the first term of the simplified form. Now we're gonna repeat this process for when K is equal to one, when K is equal to one, we get four factorial because of the binomial coefficient over four minus K, which is four minus one, that quantity factorial times K factorial which is just gonna be one factorial and that's gonna be multiplied by a squared race, the power of four minus K which is four minus one times three race to the power of K, which in this case is going to be one. Now three to the power of one is just going to give us three A squared to the power of four minus one while four minus one is just going to give us three. And using our exponential multiplication, a squared race to the power of three is going to give us a to the power of six and one factorial is just going to be one because it's just one by itself. So one factorial is just one. Now, four minus one factorial is going to give us three factorial. So rewriting this left hand side is going to give us four factorial over three factorial times one, which is three factorial times A to the power of six times three, now we can go ahead and expand four factorial because recall that any factorial is just the product of the number itself times all of its decreasing terms. So in this case here, N factorial is equal to N times N minus one times N minus two times N minus three and so on and so on. So we can rewrite four factorial as four times three times two times one. Or in this case here, we can just rewrite it as four times three factorial. And since we have a three factorial in both the numerator and denominator, we can just go ahead and cancel those out. And what we are left with is four times three times A to the power of six. And that's going to give us 12 times A to the power of six. And that's gonna be the next term in the simplified form. Next, we're repeating this process for when K is equal to two, when K is equal to two, we get four factorial over four minus two, factorial times K factorial which is two factorial times A squared race to the power of four minus K which is four minus two times three race to the power of K which is two. Now three squared is just three times three, which is going to give us nine and A squared raised to the power of four minus two. While four minus two is just going to give us two. And using our exponential multiplication. A squared race to the power of two is going to give us a to the power of four. And here two factorial can be represented as two times one. So two factorial is just going to give us two. Here we have four minus two factorial which is going to give us two factorial and rewriting. This is going to give us four factorial over two factorial times two times eight to the power of four times nine. Now we can go ahead and expand four factorial once again to try to cancel up to two factorial. Four factorial can be rewritten as four times three times two factorial. And since we have a two factorial on both the numerator and denominator, we can just go ahead and cancel them out. And what this leaves us with is four times three times nine, divided by two. And that's going to give us 54 and 54 times eight to the power of four is going to give us 54 A to the fourth. And that's our next term. Once again, we repeat this process for when K is equal to three and when Ks equal to three, we get four factorial over four minus three factorial times K factorial which is three factorial times A squared race in the power of four minus K, which is four minus three times three race to the power of K which is three. Now three cubed is just going to give us 27. So we can rewrite this as 27 and A squared race to the power of four minus 34 minus three is just going to give us one and using our exponential multiplication A squared race to the power of one is just going to give us a squared. Now three factorial can be represented as three times two times one. So we can rewrite three factorial as six and four minus three. Factorial is just going to give us one factorial, which is just represented as one and rewriting. Everything is going to give us four factorial over six times 27 times A squared. Now, we need to evaluate this four factorial and in order to do that, we're just gonna go ahead and expand it. So expanding four factorial is going to give us four times three times two times one. So four times three times two times one times 27 all of that divided by six is going to give us 108 and 108 times A squared is going to give us 108 A squared, which is the next term. And finally, we're going to evaluate this when K is equal to four. So doing this is going to give us four factorial over four minus four factorial times K factorial which is going to give us four factorial times A squared race to the power of four minus K which is four minus four times three race to the power of four. Now, this is pretty simple to evaluate because already if you notice we have a four factorial in both the numerator and denominator, so we can just go ahead and cancel those out and four minus four is going to give us zero. So this whole thing becomes zero factorial, which is just one. Now, four minus four for a squared race to the power of four minus four is just going to give us zero. And using our exponential multiplication, we get a to the power of zero, which is just going to give us one. So what we are left with is just three to the power of four and three to the power of four is going to give us 81 and that's gonna be the final term in our binomial expansion. Now, we want to go ahead and rewrite the original binomial and simplify term using all of the terms that we evaluate it for. So once again, we're going to rewrite the original binomial, which is going to be a squared plus three to the power of four. And that's going to equal to the first term, which is a to the power of eight plus our second term which is 12 A to the power of six plus the third term, which is 54 A to the power of four plus the fourth term, which is 108 A squared plus our last term which is going to be 81. And this entire thing here is going to be the original binomial in simplified form. And this means that the, the answer to this problem is going to be B So I hope this video helps you in understanding how to use the binomial theorem to rewrite a binomial in simplified form. And I'll go ahead and see you all in the next video.

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