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 finding a simplified form of the binomial. So before we jump into this, let's just quickly recall what the binomial theorem states. So the binomial theorem states that if we have the binomial in the form of A plus B raised to the power of end, then this could be rewritten as the some nation were K is equal to zero to end of the binomial coefficient and K times a race the power of n minus K times B raised to the power of K. And this is where the binomial coefficient N K is equal to N factorial over n minus K factorial times K factorial. And keep in mind that zero factorial is equal to one. So in order to do this problem, let's go ahead and first figure out what are A B and N. R. So in the binomial given to us, we have three y minus one. Race to the fourth power three Y is in the slot of A. So A is going to equal to three Y and negative one is in the slot of B. So B is going to equal to negative one. And finally, end is just the power that the binomial is raised to. So n is going to equal to four. Now writing this in the summation is going to give us the summation from one. K is equal to 0 to 4 of the binomial coefficient 40 times A which is three Y, raise the power of N which is four minus K times B which is negative one raise the power of K. So in order to find the simplified form of this binomial theorem, we want to go ahead and find all the terms from when K is equal to zero. All the way to when K is equal to four. So let's go ahead and start that. So when K is equal to zero The binomial expansion of the coefficient is going to be in factorial or in this case four factorial over four minus K which is zero factorial times K factorial which is zero factorial. And that's being multiplied by A which is three Y raised to the power of four minus K, which is four minus zero times negative one race to the power of K which is zero. Okay, so let's go ahead and simplify this. So -1 race, the power of zero is just gonna be one. Three Y. Race, the power of four minus zero is just gonna be three Y raised to the power of four and three Y. Race, the power of four is going to be three to the power four times Y to the fourth And 4 0 is just gonna be four. So we have four factorial on both the numerator and denominator and we can go ahead and cancel that out And zero factorial is just gonna be one. So what we are left with is three raised to the fourth, power times y to the fourth times one times one And that is going to equal to 81 times y to the 4th. So we found the term for when K is equal to zero. Now we need to go ahead and repeat this process for K is equal 123 and four. So when K is equal to one we get four factorial over four minus one factorial times one factorial. And that's being multiplied by three. Y raised to the power of four minus one times negative one. Race to the power of one. So negative one raised to the power of one is just going to be negative 14 minus one is going to be three and three Y race the power of three is gonna be 27 Y cubed And 4 -1 is going to be three. So we have three factorial on the denominator and one factorial is just gonna be one. Now we can go ahead and rewrite four factorial as four times three factorial. And that means that we're gonna have a three factorial on both the numerator and denominator. So we can go ahead and cancel those out. This is going to leave us with four times 27 y cubed times negative one. And that's going to give us -108 times y raised to the power of three. Now let's go ahead and repeat 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 two factorial times three y race of power of four minus two times negative one raised to the power of two negative one squared is just gonna be positive 14 minus two is going to give us two and three Y. Race the power of two is gonna give us nine y squared 4 -2 factorial is gonna give us two factorial and two factorial is just gonna be two times one. So since we have two instances of two factorial that's gonna be two times two. Which is going to give us four and four factorial could be rewritten as four times three factorial. And since we have a four on both the numerator and denominator we can just go ahead and cancel those out. So what this leaves us with is three factorial times nine y squared times one and three factorial is the same thing as three times two times one. Which is just going to be six. So nine y squared times one is just gonna be nine y squared and nine times six is gonna be 54. So what this gives us is 54 Y squared. Now when K is equal to three We get four factorial over four minus three factorial times three factorial times three Y race, the power of four minus three times negative one race, the power of three, negative one race to the power of three is just going to give us negative 14 minus three is just one. So three y. Race to the power of one is just gonna be three Y. Now 4 -3 factorial is gonna give us one factorial which is just one. And four factorial could be rewritten as four times three factorial. And since we have a three factorial on both the numerator and denominator, those are just gonna cancel out. So what we're left with is four times three y times negative one. And that's gonna give us negative 12 Y. And finally when K. is equal to four, what we get is four factorial over four minus four factorial times four factorial times three Y. Race the power of four minus four times negative one race. The power of four. Now negative one race of the power of four is gonna give us positive minus four is zero and three Y. Race of the power of zero is just gonna be one. Now we have a four factorial in both the numerator and denominator here. So we can just go ahead and cancel those out. And 4 -4 factorial is just gonna be zero factorial which is one. So what we are left with is one times one times one which is just going to be one. And that means that the simplified version of the binomial three y minus one race. The fourth power is equal to 81 Y. To the fourth minus 108 Y cubed plus 54 Y squared minus 12 Y plus one. And that's gonna be the solution to our problem. So I hope this video helps you in understanding how to use the binomial theorem to find a simplified version of a binomial, and I'll go ahead and see you all in the next video.

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