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 minus 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 minus two three times or you can use the binomial theorem to help you to expand this expression we're gonna 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 and then let's place two other ones one plus one is two at the end you will always have a one one plus two is three one plus three is four three plus three is six hopefully by now you see a pattern one plus four is five four plus six is ten and so forth now notice that we have an exponent of three so we want to use this rule where the second number is three so the coefficients that will be useful to us are one three three and one the coefficient for the first term it goes in order from left to right it's one and then we're gonna have the first letter x raised to the third power and then the second part of the binomial which is negative two 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 3 this particular number now the next term will have the second coefficient of 3. 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 negative two is now two and then the last term will have a coefficient of one x will have a power of zero and negative two 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 0 power is 1. so we could 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 6 x squared negative two squared that's negative two times negative two that's four times three is twelve so we have twelve x and now the last term negative two to the third power is negative eight and x to the zero is one so this is simply negative eight now let's confirm the answer by foiling x minus two three times so what we're gonna do is foil the first two terms x times x is x squared x times negative two is negative two x negative two times x is also negative two x and finally negative two times negative two is positive four 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 two is negative two x squared and then we have negative four x times x which is negative four x squared and negative 4x times negative 2 that's positive 8x and then 4 times x is 4x and finally we have four times negative two which is negative eight so now let's combine like terms negative two x squared minus four x 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 fourth 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 one two one one plus two is three one plus three is four three plus three is six 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 one four six four one 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 3y we're going to raise it to the zero power and then the next coefficient is going to be 4 times 2x and then it's going to be in descendant 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 0 power times three y raised to the fourth power so now let's simplify the expression that we have so anything raised to the zero power is one now what is 2x raised to the fourth power so 2x times 2x times 2x times 2x four 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 four it's going to be four times 2 to the third power times 3 to the first 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 gonna be four times two to the third two to the third power is eight and eight times three is twenty four four times twenty four is ninety six so the coefficient of the second term is ninety six 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 six times two squared times three squared two squared is four three squared is nine and six times four is twenty four twenty four times nine is two sixteen so that's the coefficient of the third term and it's going to be x squared y squared now what about the next one so we have four times two to the first power times three to the third four times two is eight three to the third power that's three times 3 times 3 that's 27 and 8 times 27 is 216. so this is going to be 216 x to the first power y to the third and then the last term we don't have to worry about two x to the zero that's equal to one so we have simply three 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 decrease in order and the exponent is associated with y is written in increasing order so you should always see that so let's say if you have a question that asks you what is the fourth term the fourth term is 216 x y cubed 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 minus 4y raised to the sixth 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 notes one one one and then one two one one three three one one four six four one four plus one is five four plus six is ten one plus five is six five plus ten is fifteen ten plus ten is twenty and the right side is the same as the left side since we have a six for the exponent we wanna 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 times three x raised to the sixth power times negative four y raised to the zero power plus the next one is going to be 6 times 3x raised to the fifth power times negative 4y raised to the first power and then plus 15 times 3x raised to the fourth power times negative 4y raised to the second power the next one is 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 six 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 sixth power so now let's simplify what we have so negative four y to the zero we could ignore that three to the sixth power is a big number it's 729 times x to the sixth and then we have six times three to the fifth power times negative four three to the fifth power is 243 times six that's 1458 times negative four so you should have negative 5832 x to the fifth power y to the first 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 fourth is 81 times 15 which is 12 15 times 4 squared so you should have 19 440 x to the fourth power y to the second power now let's try the next one so we have 3 to the third which is 27 times 20 that's 540 times 4 to the third power which is thirty four thousand five hundred sixty but it's going to be negative thirty four five sixty x to the third and then uh y to the third now for the next one 4 to the fourth power is 256 times 3 squared or 9 that's 2304 times 15 and this is going to be thousand positive hundred sixty x to the second power y to the fourth power now the next one is four to the fifth which is a thousand twenty four times three and times six this is going to be negative eighteen thousand four hundred thirty two x to the first power y to the fifth power and finally the last one four to the sixth power is four thousand ninety six so it's going to be positive four thousand ninety-six y to the sixth power three x to zero is one 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 sixth power before you can use the equation you need to understand combinations and how it relates to a 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 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 divided by n minus r factorial times r factorial the first value is five c zero 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 a value for it so n is five n minus r that's five minus two that's gonna be three and r is two so five factorial is basically five times four times three times two times one three factorial is three times two times one and two factorial is two times one so we can cancel three two and one and so we have five times four which is twenty two times one is two 20 divided by 2 is 10. so it gives us this number let's try another example let's find six c five so the first one is six c zero the next one is six c one this is six c two and then six c three and six c four and then 6c5 so 6 c5 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 6 c 4. 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 six factorial is going to be six times five times four to one which we can leave it as four factorial and two factorial is simply two times one 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 minus r times 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 minus 4y raised to the sixth power and in the last example we wanted to find the fourth term now keep in mind we knew that it was negative 34 560 x cubed y cubed you can rewind a video and you can confirm 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 times a a is basically three x which is raised to the n minus r which is uh six minus three times b which is negative four y and that's raised to the r power which is three and six c three that's six factorial divided by n minus r which is six minus three factorial or three factorial times r factorial which is also three factorial and six minus three is three so now let's figure out what this is equal to six factorial is going to be equal to six times five times four times three factorial and three factorial is three times two times one times the other three factorial so we can cancel these two six times well three times two is six so that cancels leaving behind five times four which is twenty so let's make some space so we have 20 times 3x to the third power is 27 x 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 thirty four thousand five hundred sixty 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 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 times a raised to the n minus r times b raised to the r so n is six 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 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 6 c4 a is still three x raised to the n minus r or six minus four b is negative four y raised to the r power or fourth power so six c four that's n factorial or six factorial divided by six minus four factorial which is two factorial times four factorial and then this is going to be three x raised to the second power times negative four y raised to the fourth power six factorial is six times five times four factorial two factorial simply just two so we could cancel the four factorial six times five is thirty divided by two is fifteen three squared is nine and negative four to the fourth power is positive two 256 so 15 times 9 which is 135 times 256 that's 34 and we have an x squared and a y 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 one