Understanding Binomial Expansion Techniques

In this video, we're going to focus on FOIL-in binomial expressions using Pascal's triangle, and also how to find the coefficient of, let's say, the 4th term or the 7th term, and things like that. So let's say if we have the expression x minus 2 raised to the 3rd power. How can we FOIA this expression? Now there's two ways you can do this. You can multiply x minus 2 three times or you can use the binomial theorem to help you to expand this expression. We're going to do it both ways but first let's use the binomial theorem and then we'll confirm the answer by actually FOIAing these three terms. So you need to be familiar with Pascal's triangle. So let's start with a 1, and then let's place two other ones. 1 plus 1 is 2. At the end, you will always have a 1. 1 plus 2 is 3. 1 plus 3 is 4, 3 plus 3 is 6. Hopefully by now you see a pattern. 1 plus 4 is 5, 4 plus 6 is 10, and so forth. Now, notice that we have an exponent of 3. So we want to use this row, where the second number is 3. So the coefficients that will be useful to us are 1, 3, 3, and 1. The coefficient for the first term, it goes in order from left to right. It's 1. And then we're going to have the first letter, x, raised to the third power. Then the second part of the binomial which is negative 2 and it's going to be raised to the 0 power Now these two exponents must always add to 10. I mean not 10, but must always add to 3 this particular number Now the next term will have the second coefficient of 3 Now, this exponent will decrease to 0, and this exponent will increase eventually to 3. So, this is going to be 2, and the next one is going to go up to 1. Now, the third term will have a coefficient of 3. And then the exponent... associated with x is 1, and the exponent associated with negative 2 is now 2. And then the last term will have a coefficient of 1. x will have a power of 0, and negative 2 will now have a power of 3. So that's how you can use the binomial theorem. But now let's simplify the expression that we have. Anything raised to the 0 power is 1, so we can ignore this value. So the first term is simply x cubed. Now what about the next one? 3 times negative 2 is negative 6. So it's negative 6x squared. Negative 2 squared, that's negative 2 times negative 2, that's 4, times 3 is 12. So we have 12x. And now the last term, negative 2 to the third power is negative 8. And x to the 0 is 1, so this is simply negative 8. Now let's confirm the answer by FOILing x-2 three times. So what we're going to do is FOIL the first two terms. x times x is x squared. x times negative 2 is negative 2x. Negative 2 times x is also negative 2x. And finally, negative 2 times negative 2 is positive 4. Now we can combine the middle terms. Negative 2x minus 2x is negative 4x. So now let's multiply the trinomial by the binomial. So x squared times x is x to the third power. And x squared times negative 2. is negative 2x squared and then we have negative 4x times x which is negative 4x squared and negative 4x times negative 2 that's positive 8x and then 4 times x is 4x And finally, we have 4 times negative 2, which is negative 8. So now, let's combine like terms. Negative 2x squared minus 4x squared adds up to negative 6x squared. And 8x plus 4x adds to 12x. So we do indeed get the same expression. So using the binomial theorem, it can help you to FOIL Difficult expressions. So let's say if you want to expand this expression 2x plus 3y raised to the 4th power. You can multiply it four times, or you can use the binomial theorem to expand it for you, which is a lot easier. So let's go ahead and do this example. Feel free to pause the video and work on it. Now I'm going to recreate the Pascal triangle. So it was 1, 2, 1. 1 plus 2 is 3. 1 plus 3 is 4. 3 plus 3 is 6. And we only need to go to this row, where the second number is a 4, since the exponent is 4. So the coefficients are 1, 4, 6, 4, 1. So the first term is going to have a coefficient of 1. And this part, 2x, we're going to raise it to the 4th power. And the second part, 3y, we're going to raise it to the 0 power. And then the next coefficient is going to be 4. times 2x, and then it's going to be in descending order, so it's going to be to the third power, times 3y to the first power. And then the next term is going to have a coefficient of 6, times 2x raised to the second power, times 3y raised to the second power. plus 4 times 2x raised to the first power, and then 3y raised to the third power. And then the last term is going to have the last coefficient of 1 times 2x raised to the zero power. times 3y raised to the fourth power. So now let's simplify the expression that we have. So anything raised to the zero power is 1. Now what is 2x raised to the fourth power? So 2x times 2x times 2x times 2x, 4 times. 2 to the fourth is 16, so it's going to be 16x to the fourth. Now what about the next term? What is the coefficient of the next term? The coefficient is not simply 4. It's going to be 4 times 2 to the 3rd power times 3 to the 1st power. You have to incorporate these numbers. Because sometimes you might get a question that asks you, Hey, what is the coefficient of the 2nd term? And it's not simply 4. In this case, it's going to be 4 times 2 to the 3rd. 2 to the 3rd power is 8. And 8 times 3 is 24. 4 times 24 is 96. So the coefficient of the second term is 96. And let's not forget the y variable. We do have a y here. Now what is the coefficient of the third term? So it's going to be 6 times 2 squared times 3 squared. 2 squared is 4. 3 squared is 9. And 6 times 4 is 24. 24 times 9 is 216. So that's the coefficient. The third term and it's going to be x squared y squared Now what about the next one? So we have 4 times 2 to the first power times 3 to the third. 4 times 2 is 8. 3 to the third power, that's 3 times 3 times 3, that's 27. And 8 times 27 is 216. So this is going to be 216x to the first power. y to the third. And then the last term, we don't have to worry about 2x to the 0, that's equal to 1. So we have simply 3 to the fourth power, which is 81, times y to the fourth. So as you can see, all of the exponents associated with x is written in decreasing order and the exponents associated with y is written in increasing order. So you should always see that. So let's say if you have a question, I ask you, what is the fourth term? The fourth term is 216xy cubed. If they ask you for the coefficient of the fourth term, it's 216. Now, here's another question for you. Let's say if we have 3x minus 4y raised to the 6th power. Now, what I want you to do is find the fourth term and also determine the coefficient of the fourth term. How would you do it? So we're going to do it two ways. Once again, we're going to use the binomial theorem to expand it and just simply find the fourth term. It's going to take some time, but it's going to give us the right answer. And then we're going to confirm. that answer using an equation that will help us to get the fourth term only. Instead of writing all six terms, it's going to give us the fourth term directly. It can give us any term directly if we want it. So first let's create the triangle. So we know it's 1. 1, 1, and then 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1. 4 plus 1 is 5, 4 plus 6 is 10, 1 plus 5 is 6. 5 plus 10 is 15, 10 plus 10 is 20, and the right side is the same as the left side. Since we have a 6 for the exponent, we want to start using a row that has a 6 as the second term. So let's begin. So for the first term it's going to be 1 times 3x raised to the 6th power times negative 4y raised to the 0 power. Plus, the next one is going to be 6 times 3x raised to the 5th power times negative 4y raised to the 1st power. and then plus 15 times 3x raised to the fourth power times negative 4y raised to the second power. The next one's going to be 20 times 3x to the third power. times negative 4y to the third power. And then it's going to be 15, 3x squared, negative 4y to the second power. Actually, to the fourth power. The exponents must add to 6 in this particular problem. And then the last one is going to be 1 times 3x raised to the 0 power times negative 4y raised to the 6th power. So now let's simplify what we have. So negative 4y to the 0, we can ignore that. 3 to the 6th power is a big number, it's 729, times x to the 6th. And then we have 6 times 3 to the 5th power times negative 4. 3 to the 5th power is 243 times 6. That's 1458 times negative 4. So you should have negative 5832. X to the 5th power. Y to the 1st power. So the signs are going to alternate between positive and negative. Now for the next term, it's going to be positive. 3 to the 4th is 81, times 15, which is 1215, times 4 squared. So you should have 19,440, x to the 4th power, y to the 2nd power. Now let's try the next one. so we have three to the third which is 27 times 20 that's 540 times 4 to the third power which is 34,560 but it's going to be negative 34 560 X to the third and then Y to the third Now for the next one, 4 to the 4th power is 256 times 3 squared or 9. That's 2304 times 15. And this is going to be positive 34,560. X to the 2nd power, Y to the 4th power. Now the next one is 4 to the 5th, which is 1024 times 3 and times 6. This is going to be negative 18,432. X to the 1st power, Y to the 5th power. And finally, the last one. 4 to the 6th power is 4096. So it's going to be positive 4096, Y to the 6th power. 3x to 0 is 1. Now our goal is to find the fourth term. So this is the first term, this is the second, this is the third, and here is the fourth term. It's negative 34,560 x cubed y cubed. So somehow we need to get this value. Let's see if there's an easier way to get that answer. Just make sure you write it down. So the original expression was 3x minus 4y raised to the 6th power. Before you can use the equation, you need to understand combinations and how it relates to Pascal's triangle. So I'm going to redraw the triangle up to the sixth level. So let's say if we want to find the third number in Pascal's triangle in this particular row where the second value is 5. Let's call this the first row. Let's call this one the 0 row. So this will be row number 2. So therefore the fifth row would be this one. So if you want to get this value, you need to use this expression ncr. It's a combination. It's equal to n factorial divided by n minus r factorial times r factorial. The first value is 5c0. The second value is 5c1. The third value is 5c2. This is 5c3, and then 5c4, 5c5. So 5c2 should give us 10. So let's find the value for it. So n is 5. n minus r, that's 5 minus 2, that's going to be 3, and r is 2. So 5 factorial is basically 5 times 4 times 3 times 2 times 1. 3 factorial is 3 times 2 times 1. and 2 factorial is 2 times 1. So we can cancel 3, 2, and 1, and so we have 5 times 4, which is 20. 2 times 1 is 2. 20 divided by 2 is 10. So it gives us this number. Let's try another example. Let's find 6, C5. So the first one is 6, C0. The next one is 6, C1. This is 6, C2. And then 6, C3. And 6, C4. And then 6, C5. So 6c5 should give us a value of 6. So let's go ahead and calculate it. This is going to be 6 factorial divided by 6 minus 5, which is 1 factorial, times 5 factorial. So 6 factorial is 6 times... 5 times 4 times 3 times 2 times 1. But instead of writing 5 times 4 all the way to 1, we can leave it as 5 factorial. 1 factorial is simply 1. And notice that we can cancel the 5 factorial, which is going to give us 6. Now, let's calculate 6C4. Let's see that it's going to give us... Let's prove that's going to give us 15. So it's going to be 6 factorial divided by n minus r, which is 6 minus 4, or 2 factorial, and r is 4. So 6 factorial is going to be 6 times 5 times 4 to 1, which we can leave it as 4 factorial. And 2 factorial is simply 2 times 1. And then we can cancel the 4 factorials. 6 times 5 is 30. And 30 divided by 2 is 15. So it gives us this particular value. So as you can see, you can use combinations to find any value in Pascal's triangle. Now this is going to help us to find any term whenever we need to expand it. So if we have the expression a plus b raised to the nth power, and we wish to find a certain term, we can use this expression. ncr times a raised to the n-r. Times B raised to the R power where R is basically the term minus 1 So let's go back to our original problem, which was 3x minus 4y raised to the 6th power. And in the last example, we wanted to find the fourth term. Now keep in mind, we knew that it was negative 34,560x cubed y cubed. You can rewind the video and you can confirm it if you want to. But let's use this expression to get this answer directly. So, we need to find out what is n and what is r. n is basically the exponent that you see here. So, n is equal to 6. Now, what about r? Well, if you're looking for the fourth term, r is going to be one less than the fourth term, so r is 3. So with that, we could find the term and its coefficient. So it's going to be nCr, which is 6C3, times a. a is basically 3x. Which is raised to the n minus r which is 6 minus 3? times B which is negative 4y and that's raised to the r power which is 3 and 6 C3 that's 6 factorial divided by n minus r which is 6 minus 3 factorial or 3 factorial times r factorial which is also 3 factorial and 6 minus 3 is 3. So now let's figure out what this is equal to. 6 factorial is going to be equal to 6 times 5 times 4 times 3 factorial. And 3 factorial... is 3 times 2 times 1 times the other 3 factorial. So we can cancel these two. 6 times, well, 3 times 2 is 6, so that cancels, leaving behind 5 times 4, which is 20. So let's make some space. So we have 20 times 3x to the third power is 27x cubed. Negative 4 to the third power is negative 64 times y cubed. Now, if we multiply 20 times 27 times negative 64, this is going to give us... negative 34,560 and then you can see it's x cubed y cubed so this is the fourth term and the coefficient of the fourth term is negative 34,560 now let's find the value of the fifth term using the same expression Now, if you rewind the video, you'll see that the fifth term is 34,560 x squared y to the fourth power. So go ahead and pause the video, see if you can find the fifth term using the equation that I just showed you. So let's begin. We know it's going to be nCr times a raised to the n minus r times b raised to the r. So n is 6. And r is basically the term minus 1. Since we're looking for the fifth term, r is going to be 5 minus 1, which is 4. So now let's plug in everything, and let's see if we're going to get this answer. So ncr, that's going to be 6c4. a is still 3x, raised to the n minus r, or 6 minus 4. b. is negative 4y raised to the r power or fourth power. So 6c4, that's n factorial or 6 factorial divided by 6 minus 4 factorial, which is 2 factorial, times 4 factorial. And then this is going to be 3x raised to the second power times negative 4y raised to the fourth power. 6 factorial is 6 times 5 times 4 factorial. 2 factorial is simply just 2, so we can cancel the 4 factorial. 6 times 5 is 30, divided by 2 is 15. 3 squared is 9, and negative 4 to the 4th power is positive 256. So 15 times 9, which is 135, times 256, that's 34,560. And we have an x squared and a y cubed to the fourth. So as you can see, it gives us the same answer. So now you know how to find any term and its coefficient by using this expression. So just remember, n is basically the exponent, and r is whatever term you have, or whatever term you're looking for, minus 1.

You can multiply x minus 2 three times or you can use the binomial theorem to help you to expand this expression. We're going to do it both ways but first let's use the binomial theorem and then we'll confirm the answer by actually FOIAing these three terms. So you need to be familiar with Pascal's triangle.

So let's start with a 1, and then let's place two other ones. 1 plus 1 is 2. At the end, you will always have a 1. 1 plus 2 is 3. 1 plus 3 is 4, 3 plus 3 is 6. Hopefully by now you see a pattern. 1 plus 4 is 5, 4 plus 6 is 10, and so forth.

Now, notice that we have an exponent of 3. So we want to use this row, where the second number is 3. So the coefficients that will be useful to us are 1, 3, 3, and 1. The coefficient for the first term, it goes in order from left to right. It's 1. And then we're going to have the first letter, x, raised to the third power. Then the second part of the binomial which is negative 2 and it's going to be raised to the 0 power Now these two exponents must always add to 10. I mean not 10, but must always add to 3 this particular number Now the next term will have the second coefficient of 3 Now, this exponent will decrease to 0, and this exponent will increase eventually to 3. So, this is going to be 2, and the next one is going to go up to 1. Now, the third term will have a coefficient of 3. And then the exponent... associated with x is 1, and the exponent associated with negative 2 is now 2. And then the last term will have a coefficient of 1. x will have a power of 0, and negative 2 will now have a power of 3. So that's how you can use the binomial theorem. But now let's simplify the expression that we have.

Anything raised to the 0 power is 1, so we can ignore this value. So the first term is simply x cubed. Now what about the next one? 3 times negative 2 is negative 6. So it's negative 6x squared. Negative 2 squared, that's negative 2 times negative 2, that's 4, times 3 is 12. So we have 12x.

And now the last term, negative 2 to the third power is negative 8. And x to the 0 is 1, so this is simply negative 8. Now let's confirm the answer by FOILing x-2 three times. So what we're going to do is FOIL the first two terms. x times x is x squared.

x times negative 2 is negative 2x. Negative 2 times x is also negative 2x. And finally, negative 2 times negative 2 is positive 4. Now we can combine the middle terms. Negative 2x minus 2x is negative 4x.

So now let's multiply the trinomial by the binomial. So x squared times x is x to the third power. And x squared times negative 2. is negative 2x squared and then we have negative 4x times x which is negative 4x squared and negative 4x times negative 2 that's positive 8x and then 4 times x is 4x And finally, we have 4 times negative 2, which is negative 8. So now, let's combine like terms.

Negative 2x squared minus 4x squared adds up to negative 6x squared. And 8x plus 4x adds to 12x. So we do indeed get the same expression. So using the binomial theorem, it can help you to FOIL Difficult expressions. So let's say if you want to expand this expression 2x plus 3y raised to the 4th power.

You can multiply it four times, or you can use the binomial theorem to expand it for you, which is a lot easier. So let's go ahead and do this example. Feel free to pause the video and work on it.

Now I'm going to recreate the Pascal triangle. So it was 1, 2, 1. 1 plus 2 is 3. 1 plus 3 is 4. 3 plus 3 is 6. And we only need to go to this row, where the second number is a 4, since the exponent is 4. So the coefficients are 1, 4, 6, 4, 1. So the first term is going to have a coefficient of 1. And this part, 2x, we're going to raise it to the 4th power. And the second part, 3y, we're going to raise it to the 0 power.

And then the next coefficient is going to be 4. times 2x, and then it's going to be in descending order, so it's going to be to the third power, times 3y to the first power. And then the next term is going to have a coefficient of 6, times 2x raised to the second power, times 3y raised to the second power. plus 4 times 2x raised to the first power, and then 3y raised to the third power.

And then the last term is going to have the last coefficient of 1 times 2x raised to the zero power. times 3y raised to the fourth power. So now let's simplify the expression that we have.

So anything raised to the zero power is 1. Now what is 2x raised to the fourth power? So 2x times 2x times 2x times 2x, 4 times. 2 to the fourth is 16, so it's going to be 16x to the fourth.

Now what about the next term? What is the coefficient of the next term? The coefficient is not simply 4. It's going to be 4 times 2 to the 3rd power times 3 to the 1st power.

You have to incorporate these numbers. Because sometimes you might get a question that asks you, Hey, what is the coefficient of the 2nd term? And it's not simply 4. In this case, it's going to be 4 times 2 to the 3rd. 2 to the 3rd power is 8. And 8 times 3 is 24. 4 times 24 is 96. So the coefficient of the second term is 96. And let's not forget the y variable. We do have a y here.

Now what is the coefficient of the third term? So it's going to be 6 times 2 squared times 3 squared. 2 squared is 4. 3 squared is 9. And 6 times 4 is 24. 24 times 9 is 216. So that's the coefficient. The third term and it's going to be x squared y squared Now what about the next one? So we have 4 times 2 to the first power times 3 to the third.

4 times 2 is 8. 3 to the third power, that's 3 times 3 times 3, that's 27. And 8 times 27 is 216. So this is going to be 216x to the first power. y to the third. And then the last term, we don't have to worry about 2x to the 0, that's equal to 1. So we have simply 3 to the fourth power, which is 81, times y to the fourth. So as you can see, all of the exponents associated with x is written in decreasing order and the exponents associated with y is written in increasing order. So you should always see that.

So let's say if you have a question, I ask you, what is the fourth term? The fourth term is 216xy cubed. If they ask you for the coefficient of the fourth term, it's 216. Now, here's another question for you. Let's say if we have 3x minus 4y raised to the 6th power. Now, what I want you to do is find the fourth term and also determine the coefficient of the fourth term.

How would you do it? So we're going to do it two ways. Once again, we're going to use the binomial theorem to expand it and just simply find the fourth term.

It's going to take some time, but it's going to give us the right answer. And then we're going to confirm. that answer using an equation that will help us to get the fourth term only. Instead of writing all six terms, it's going to give us the fourth term directly.

It can give us any term directly if we want it. So first let's create the triangle. So we know it's 1. 1, 1, and then 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1. 4 plus 1 is 5, 4 plus 6 is 10, 1 plus 5 is 6. 5 plus 10 is 15, 10 plus 10 is 20, and the right side is the same as the left side. Since we have a 6 for the exponent, we want to start using a row that has a 6 as the second term. So let's begin.

So for the first term it's going to be 1 times 3x raised to the 6th power times negative 4y raised to the 0 power. Plus, the next one is going to be 6 times 3x raised to the 5th power times negative 4y raised to the 1st power. and then plus 15 times 3x raised to the fourth power times negative 4y raised to the second power. The next one's going to be 20 times 3x to the third power.

times negative 4y to the third power. And then it's going to be 15, 3x squared, negative 4y to the second power. Actually, to the fourth power.

The exponents must add to 6 in this particular problem. And then the last one is going to be 1 times 3x raised to the 0 power times negative 4y raised to the 6th power. So now let's simplify what we have.

So negative 4y to the 0, we can ignore that. 3 to the 6th power is a big number, it's 729, times x to the 6th. And then we have 6 times 3 to the 5th power times negative 4. 3 to the 5th power is 243 times 6. That's 1458 times negative 4. So you should have negative 5832. X to the 5th power. Y to the 1st power. So the signs are going to alternate between positive and negative.

Now for the next term, it's going to be positive. 3 to the 4th is 81, times 15, which is 1215, times 4 squared. So you should have 19,440, x to the 4th power, y to the 2nd power. Now let's try the next one. so we have three to the third which is 27 times 20 that's 540 times 4 to the third power which is 34,560 but it's going to be negative 34 560 X to the third and then Y to the third Now for the next one, 4 to the 4th power is 256 times 3 squared or 9. That's 2304 times 15. And this is going to be positive 34,560.

X to the 2nd power, Y to the 4th power. Now the next one is 4 to the 5th, which is 1024 times 3 and times 6. This is going to be negative 18,432. X to the 1st power, Y to the 5th power.

And finally, the last one. 4 to the 6th power is 4096. So it's going to be positive 4096, Y to the 6th power. 3x to 0 is 1. Now our goal is to find the fourth term.

So this is the first term, this is the second, this is the third, and here is the fourth term. It's negative 34,560 x cubed y cubed. So somehow we need to get this value.

Let's see if there's an easier way to get that answer. Just make sure you write it down. So the original expression was 3x minus 4y raised to the 6th power. Before you can use the equation, you need to understand combinations and how it relates to Pascal's triangle. So I'm going to redraw the triangle up to the sixth level.

So let's say if we want to find the third number in Pascal's triangle in this particular row where the second value is 5. Let's call this the first row. Let's call this one the 0 row. So this will be row number 2. So therefore the fifth row would be this one. So if you want to get this value, you need to use this expression ncr.

It's a combination. It's equal to n factorial divided by n minus r factorial times r factorial. The first value is 5c0. The second value is 5c1. The third value is 5c2.

This is 5c3, and then 5c4, 5c5. So 5c2 should give us 10. So let's find the value for it. So n is 5. n minus r, that's 5 minus 2, that's going to be 3, and r is 2. So 5 factorial is basically 5 times 4 times 3 times 2 times 1. 3 factorial is 3 times 2 times 1. and 2 factorial is 2 times 1. So we can cancel 3, 2, and 1, and so we have 5 times 4, which is 20. 2 times 1 is 2. 20 divided by 2 is 10. So it gives us this number. Let's try another example.

Let's find 6, C5. So the first one is 6, C0. The next one is 6, C1. This is 6, C2.

And then 6, C3. And 6, C4. And then 6, C5. So 6c5 should give us a value of 6. So let's go ahead and calculate it. This is going to be 6 factorial divided by 6 minus 5, which is 1 factorial, times 5 factorial.

So 6 factorial is 6 times... 5 times 4 times 3 times 2 times 1. But instead of writing 5 times 4 all the way to 1, we can leave it as 5 factorial. 1 factorial is simply 1. And notice that we can cancel the 5 factorial, which is going to give us 6. Now, let's calculate 6C4. Let's see that it's going to give us... Let's prove that's going to give us 15. So it's going to be 6 factorial divided by n minus r, which is 6 minus 4, or 2 factorial, and r is 4. So 6 factorial is going to be 6 times 5 times 4 to 1, which we can leave it as 4 factorial.

And 2 factorial is simply 2 times 1. And then we can cancel the 4 factorials. 6 times 5 is 30. And 30 divided by 2 is 15. So it gives us this particular value. So as you can see, you can use combinations to find any value in Pascal's triangle.

Now this is going to help us to find any term whenever we need to expand it. So if we have the expression a plus b raised to the nth power, and we wish to find a certain term, we can use this expression. ncr times a raised to the n-r.

Times B raised to the R power where R is basically the term minus 1 So let's go back to our original problem, which was 3x minus 4y raised to the 6th power. And in the last example, we wanted to find the fourth term. Now keep in mind, we knew that it was negative 34,560x cubed y cubed.

You can rewind the video and you can confirm it if you want to. But let's use this expression to get this answer directly. So, we need to find out what is n and what is r. n is basically the exponent that you see here. So, n is equal to 6. Now, what about r?

Well, if you're looking for the fourth term, r is going to be one less than the fourth term, so r is 3. So with that, we could find the term and its coefficient. So it's going to be nCr, which is 6C3, times a. a is basically 3x. Which is raised to the n minus r which is 6 minus 3?

times B which is negative 4y and that's raised to the r power which is 3 and 6 C3 that's 6 factorial divided by n minus r which is 6 minus 3 factorial or 3 factorial times r factorial which is also 3 factorial and 6 minus 3 is 3. So now let's figure out what this is equal to. 6 factorial is going to be equal to 6 times 5 times 4 times 3 factorial. And 3 factorial...

is 3 times 2 times 1 times the other 3 factorial. So we can cancel these two. 6 times, well, 3 times 2 is 6, so that cancels, leaving behind 5 times 4, which is 20. So let's make some space. So we have 20 times 3x to the third power is 27x cubed.

Negative 4 to the third power is negative 64 times y cubed. Now, if we multiply 20 times 27 times negative 64, this is going to give us... negative 34,560 and then you can see it's x cubed y cubed so this is the fourth term and the coefficient of the fourth term is negative 34,560 now let's find the value of the fifth term using the same expression Now, if you rewind the video, you'll see that the fifth term is 34,560 x squared y to the fourth power.

So go ahead and pause the video, see if you can find the fifth term using the equation that I just showed you. So let's begin. We know it's going to be nCr times a raised to the n minus r times b raised to the r.

So n is 6. And r is basically the term minus 1. Since we're looking for the fifth term, r is going to be 5 minus 1, which is 4. So now let's plug in everything, and let's see if we're going to get this answer. So ncr, that's going to be 6c4. a is still 3x, raised to the n minus r, or 6 minus 4. b. is negative 4y raised to the r power or fourth power.

So 6c4, that's n factorial or 6 factorial divided by 6 minus 4 factorial, which is 2 factorial, times 4 factorial. And then this is going to be 3x raised to the second power times negative 4y raised to the fourth power. 6 factorial is 6 times 5 times 4 factorial.

2 factorial is simply just 2, so we can cancel the 4 factorial. 6 times 5 is 30, divided by 2 is 15. 3 squared is 9, and negative 4 to the 4th power is positive 256. So 15 times 9, which is 135, times 256, that's 34,560. And we have an x squared and a y cubed to the fourth.

So as you can see, it gives us the same answer. So now you know how to find any term and its coefficient by using this expression. So just remember, n is basically the exponent, and r is whatever term you have, or whatever term you're looking for, minus 1.

Transcript for:Understanding Binomial Expansion Techniques

Transcript for:
Understanding Binomial Expansion Techniques