In this video, we're going to focus on foiling binomial expressions using Pascal's triangle and also how to find the coefficient of let's say the fourth term or the seventh term and things like that. So let's say if we have the expression x - 2 raised to the third power. How can we foil this expression? Now there's two ways you can do this. You can multiply x - 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 foiling these three terms. So, you need to be familiar with Pascal's triangle. So let's start with a one and then let's place two other ones. 1 + 1 is two. At the end you will always have a 1. 1 + 2 is 3. 1 + 3 is 4. 3 + 3 is 6. Hopefully by now you see a pattern. 1 + 4 is 5. 4 + 6 is 10. and so forth. Now notice that we have an exponent of three. So we want to use this row where the second number is three. 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 one. And then we're going to have the first letter x raised to the third power and then the second part of the binomial which is -2 and it's going to be raised to the zero power. Now these two exponents must always add to 10. I mean not 10 but must always add to three this particular number. Now the next term will have the second coefficient of three. Now this exponent will decrease to zero and this exponent will increase eventually to three. So this is going to be two and the next one is going to go up to one. Now the third term will have a coefficient of three and then the exponent associated with x is one and the exponent associated with -2 is now two and then the last term will have a coefficient of one. x will have a power of zero and -2 will now have a power of three. So that's how you can use the binomial theorem. But now let's simplify uh the expression that we have. Anything raised to the zero power is one. So we could ignore this uh value. So the first term is simply x cub. Now what about the next one? 3 * -2 is -6. So it's -6 x^2 -2^2 that's -2 * -2 that's 4 * 3 is 12. So we have 12x and now the last term -2 to the 3 power is8 and x to the zero is one. So this is simply8. Now let's confirm the answer by foiling x - 2 3 times. So what we're going to do is foil the first two terms. x * x is x^2. x * -2 is -2x. -2 * x is also -2x and finally -2 * -2 is pos4. Now we can combine the middle terms -2x - 2x is -4x. So now let's multiply the tromial by the binomial. So x^2 * x is x^ the 3 power and x^2 * -2 is -2x^2. And then we have -4x * x which is -4x^2. And4x * -2 that's positive 8x. And then 4 * x is 4x. And finally we have 4 * -2 which is8. So now let's combine like terms. -2x^2 - 4x^2 adds up to -6x^2 and 8x + 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 + 3 y 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's 1 2 1. 1 + 2 is 3. 1 + 3 is 4. 3 + 3 is 6. And we only need to go to this row where the second number is a four since the exponent is four. So the coefficients are 14 6 41. So the first term is going to have a coefficient of one. And this part 2x we're going to raise it to the fourth power. And the second part 3 y we're going to raise it to the zero power. And then the next coefficient is going to be 4 * 2x and then it's going to be in descending order. So it's going to be to the third power * 3 y to the first power. And then the next term is going to have a coefficient of 6 * 2x raised to the 2 power * 3 y raised to the second power + 4 * 2x raised to the first power and then 3 y raised to the third power. And then the last term is going to have the last coefficient of 1 * 2x raised to the 0 power * 3 y raised to the fourth power. So now let's simplify the expression that we have. So anything raised to the 0 power is 1. Now what is 2x raised to the 4th power? So 2x * 2x * 2x * 2x four times 2 4th is 16. So it's going to be 16 x 4th. Now what about the next term? What is the coefficient of the next term? The coefficient is not simply four. It's going to be 4 * 2 3 power * 3^ the 1st power. You have to incorporate these numbers because sometimes you might get a question to ask you, hey, what is the coefficient of the second term? And it's not simply four. In this case, it's going to be 4 * 2 3r. 2 to the 3 power is 8. And 8 * 3 is 24. 4 * 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 * 2^ 2 * 3 2 is 4 3 2 is 9. and 6 * 4 is 24. 24 * 9 is 216. So that's the coefficient of the third term and it's going to be x^2 y^2. Now what about the next one? So we have 4 * 2 1st power * 3 3r 4 * 2 is 8 3 to the 3 power that's 3 * 3 * 3 that's 27 and 8 * 27 is 216. So this is going to be 216 x to the 1st power y the 3r 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 4th power which is 81 * y 4th. 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 they ask you what is the fourth term? The fourth term is 216 x y cub. If they ask you for the coefficient of the fourth term is 216. Now here's another question for you. Let's say if we have 3x - 4 y raised to the 6 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 uh want it. So first let's create the triangle. So we know it's one 1 and then 1 2 1 1 3 3 1 1 4 6 4 1 4 + 1 is 5 4 + 6 is 10 1 + 5 is 6 5 + 10 is 15 10 + 10 is 20. and the right side is the same as the left side. Since we have a six for the exponent, we want to start using a row that has a six as the second term. So let's begin. So for the first term, it's going to be 1 * 3x raised to the 6 power * 4 y raised to the 0 power plus the next one is going to be 6 * 3x raised to the 5th power * -4 y raised to the first power and then plus 15 * 3x raised to the 4th power * -4 y raised to the second power. The next one is going to be 20 * 3x to the 3 power * -4 y to the 3 power. And then it's going to be 15 3x^2 -4 y to the second power actually to the fourth power. The exponents must add to six in this particular problem. And then the last one, it's going to be 1 * 3x raised to the 0 power * -4 y raised to the 6 power. So now let's simplify what we have. So -4 y^ 0 we could ignore that 3 to the 6 power is a big number. It's 729 * x^ 6 and then we have 6 * 3 5th power 4 3 to the 5ifth power is 243 * 6 That's 1458 * -4. So you should have -5,832 x^ 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 * 15, which is 125 * 4^ 2. So you should have 19,440 x to the 4th power y to the 2 power. Now let's try the next one. So we have 3 which is 27 * 20 that's 540 * 4^ the 3 power which is 34,560 but it's going to be negative 34560 x to the 3r and then y 3r. Now for the next one, 4^ the 4th power is 256 * 3^ 2 or 9. That's 2304 * 15. And this is going to be positive 34,560 x to the 2 power y to the 4th power. Now the next one is 4 to the 5th which is,024 * 3 and time 6. This is going to be -18,432 x to the 1st power y to the 5th power. And finally the last one 4 to the 6 power is 496. So it's going to be positive 496 y to the 6th power. 3x to the 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 -34,560 x cub y. 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 uh 3x - 4 y raised to the 6 power. Before we can use the equation, we need to understand combinations and how it relates to a Pascal 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 five. Let's call this the first row. Let's call this one the zero row. And so this will be row number two. 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 / n - r factorial * r factorial. The first value is 5 c 0. The second value is 5 c1. The third value is 5 c2. This is 5C3 and then 5C4 5C5. So 5C2 should give us 10. So let's find a value for it. So n is 5. n - r that's 5 - 2 that's going to be three. and r is 2. So 5 factorial is basically 5 * 4 * 3 * 2 * 1. 3 factorial is 3 * 2 * 1 and 2 factorial is 2 * 1. So we can cancel 3 2 and 1. And so we have 5 * 4 which is 20. 2 * 1 is 2. 20 / 2 is 10. So it gives us this number. Let's try another example. Let's find six C5. So the first one is 6 C 0. The next one is 6 C1. This is 6 C2. and then 6 C3 and 6 C4 and then 6 C5. So 6 C5 should give us a value of six. So let's go ahead and calculate it. This is going to be 6 factorial / 6 - 5 which is 1 factorial * 5 factorial. So 6 factorial is 6 * 5 * 4 * 3 * 2 * 1. But instead of writing 5 * 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 five factorial which is going to give us six. Now let's calculate 6 C4. 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 / n minus r which is 6 - 4 or 2 factorial and r is 4. So 6 factorial is going to be 6 * 5 * 4 to 1 which we can leave it as 4 factorial and 2 factorial is simply 2 * 1 and then we can cancel the four factorials. 6 * 5 is 30 and 30 / 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 + b raised to the n power and we wish to find a certain term we can use this expression n c r * a raised to the n minus r b raised to the r power where R is basically the term minus one. So let's go back to our original problem which was 3x - 4 y raised to the 6 power. And in the last example we wanted to find the fourth term. Now keep in mind we knew that it was -34 560 x cub y cub. 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 six. 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 three. So with that we could find the term and its coefficient. So it's going to be ncr which is 6 c3 * a. a is basically 3x which is raised to the n - r which is uh 6 - 3 * b which is -4 y and that's raised to the r power which is 3. and 6 C3 that's 6 factorial / n - r which is 6 - 3 factorial or 3 factorial * r factorial which is also 3 factorial and 6 - 3 is 3. So now let's figure out what this is equal to. 6 factorial is going to be equal to 6 * 5 * 4 * 3 factorial and 3 factorial is 3 * 2 * 1 * the other 3 factorial. So we can cancel these two 6 * well 3 * 2 is 6. So that cancels leaving behind 5 * 4 which is 20. So let's make some space. So we have 20 * 3x^ the 3r power is 27 x cub -4 to the 3 power is -64 * y cub. Now if we multiply 20 * 27 * -64 this is going to give us -34,560 and then you can see it's x cub y cub. So this is the fourth term and the coefficient of the fourth term is -34560. 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^2 y^ 4th power. So go ahead and pause the video. see if you could find the fifth term using the equation that I just showed you. So, let's begin. We know it's going to be n c r * a raised to the n - r * b raised to the r. So, n is 6 and r is basically the term minus one. Since we're looking for the fifth term, r is going to be 5 minus 1, which is four. 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 6 c4. A is still 3x raised to the n minus r or 6 - 4. B is -4 y raised to the r power or 4th power. So 6 C4 that's n factorial or 6 factorial / 6 - 4 factorial which is 2 factorial * 4 factorial and then this is going to be 3x raised to the 2 power * -4 y raised to the 4th power. 6 factorial is 6 * 5 * 4 factorial. 2 factorial is simply just 2. So we could cancel the 4 factorial. 6 * 5 is 30 / 2 is 15. 3^2 is 9 and -4 to the 4th power is pos 256. So 15 * 9 which is 135 * 256 that's 34,560 and we have an x^2 and a y to the 4th. 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 one.