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. (2x+1)^3

Accepted Answer

Hello today, we're going to be using the binomial theorem to rewrite the given binomial and simplified form. Now, before we jump into this, let's just quickly recall what the binomial theorem states. The binomial theorem states that if you have a binomial in the form of a plus B race, the power of N. This can be rewritten as the some nation from one 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 where the binomial coefficient N K is equal to end factorial over n minus K factorial times K factorial. And keep in mind 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 three y minus one, raised to the power of four. Using the binomial theorem, we can let a equal to three Y. We can let b equal to negative one and we can let n equal to four. And putting this into summation form is going to give us the summation from one. K is equal to zero. All the way to four of the binomial coefficient for K times A. Which is three Y race of the power of n minus K, which is four minus K times B, which is negative one raised to the power of K. Now, in order to rewrite this in simplified form, we need to go ahead and evaluate the summation for all of its terms or in other words we're going to be evaluating it when k is equal to zero, all the way to when K is equal to four. So let's just go ahead and start that we're going to start when k is equal to zero. And we first start by expanding the binomial coefficient that's going to give us n factorial which is four factorial over and minus K, which is four minus zero this time because K is equal to zero. That quantity factorial times K factorial which is zero factorial and that's going to be multiplied by three. Y raised to the power of four minus K, which in this case is going to be four minus zero times negative one raised to the power of K, which in this case is zero. Now let's just go ahead and simplify recall that zero factorial is equal to one. So zero factorial can be rewritten as one And 0 is just going to give us four. So we have four factorial and we have a four factorial in both the numerator and denominator. So we can go ahead and cancel those out. Making this whole quantity equal to one. Any number raised to the power of zero is just going to give us one. So negative one to the zero is going to give us one and three y rates of the power of four minus zero is going to give us well four minus zero is four and we need to distribute this exponent of four to both three and y. So what we are left with is three to the power of four Times Y to the fourth and three to the power of four is going to give us 81. So what we have is 81 Why? To the 4th? And that's going to be the first term for a simplified form. Now we want to go ahead and repeat this process for when K is equal to one, so when K is equal to one, we get four factorial Over -1 factorial times K Factorial which is just one factorial times three, Y raised to the power of four minus K which is four minus one times negative one. Race to the power of one. So let's go ahead and simplify this negative one race of the power of one is just going to give us negative one. Three y race of the power of four minus one. While four minus one is going to give us three and we're going to need to distribute this exponents to both Y and three. So that's going to give us three cubed times y cubed. and here one factorial can just be rewritten as one And 4 -1 factorial is going to give us three factorial. So what we have right now is four factorial Over three factorial times one which is going to give us three factorial times three cubed times y cubed, All of that times -1 and four factorial can be rewritten as four times three times two times one. As for example and factorial is just the product of all of itself and all of its decreasing terms. So n factorial can be represented as end times and minus one times n minus two times n minus three and so on and so on. So four factorial can be rewritten as four times three times two times one. Or in this case here to help us get rid of this three factorial, we can 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 this out. And what we have is four times three cubed times negative one. And that's going to give us negative 108 times Y to the third. So we have negative 108 y to the third. And that's going to be our next term. Now, once again, we're going to repeat this process for when K is equal to two. So when K is equal to two, we have four factorial Over -2 factorial times K factorial which is two factorial times three Y race of the power of four minus k which is going to be four minus two times negative one. Race of the power of K which is two. Now, any number squared is always going to produce a positive value. So negative one squared is going to give us positive one, three Y raised to the power of 4 -2 while 4 -2 is just going to give us two and once again we have to distribute this exponent to each term in three Y. And what that gives us is three squared times y squared. Now here two factorial is the same thing as two times one. So what we have is to 4 -2. Factorial is going to give us two factorial And what we have is four factorial over two factorial times two times three squared times y squared times one and we can go ahead and expand four factorial to be four times three times two and that's going to give us four times three times two factorial. And since we have a two factorial in both the numerator and denominator we can just go ahead and cancel them out. And what we have is four times three times three squared times one. All of that divided by two. That's going to give us 54 54 times y squared is going to give us 54 Y squared. That's going to be our next turn. Once again we repeat this for when K is equal to three And when K is equal to three we get four factorial over 4 - factorial times three factorial times three Y. Race. The power of four minus K which is four minus three times negative one. Race the power of K which is three now negative one to the power of the three is going to be the same thing as negative one, times negative one times negative one. And that's going to give us negative one. Four minus three is just going to give us one and three Y raise to the power of one is just going to be three Y. Three factorial can be evaluated as three times two times one and that's going to give us six And 4 -3. Factorial is going to be one factorial which just gives us one. So what we have is four factorial over one times six which is six times three Y times negative one and four factorial can be rewritten As four times 3 times two times 1. Now, four times three times two times one times three times negative one. All of that divided by six Is going to give us -12. And so we have negative 12 times y which is going to give us -12 Y. And that's going to be the next term. And finally we're going to evaluate this when K is equal to four and when K is equal to four we have four factorial over 4 -4 factorial times four factorial times three Y raise the power of four minus K which is four minus four, raise the power of negative one to the power four. Now here four factorial is in both the numerator and denominator. So we can just go ahead and cancel those out and four minus four is just going to give us zero and zero factorial is going to be one For three. Y raise to the power of 4 -4. 4 -4 is going to give us zero and any number raised to the power of zero is just going to give us one. And finally we have negative one race to the power of four. That's negative one times itself four times which is going to give us positive one. So what we have is one times one times one, which is just going to give us one for the final term. So now what we want to go ahead and do is add up all of the terms together. And that's gonna be the simplified form of the original binomial. Now recall that the original binomial was three y minus one raised to the power of four and that's going to equal to the first term, which is 81 y to the fourth minus the second term, which is 100 and eight Y cubed plus our third term which is 54 y squared minus the fourth term, which is 12 Y plus our last term, which is one. And this is going to be the completely simplified version of the original binomial through the binomial theorem And that means that the answer to this problem is going to be a so I hope this video helps you in understanding how to write the simplified version of a binomial using the binomial theorem, 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. (2x+1)^3