Term Independent and Finding Any Term

[00:00:00]and welcome to my YouTube channel. So, here today we're going to be looking at the term independent of X.

[00:00:06]So now, how do we go about that one? So, if you're new to this channel, kindly like, subscribe, and share for more.

[00:00:14]Firstly, when you're looking at the term independent, we are looking at a term without X. So, we have a famous formula, which is our T R + 1.

[00:00:25]I'll explain. So, uh should be equal to N combination of R. Then, we are going to have our A N R B to the power of R. Where if you have, let's say, if you have your 1 + X to the power of N, let's say your N is 5, you're going to have your It will be your N will be 5 combination of R, which you don't know. Then, you have your A.

[00:00:54]Then, you have your N. It will be N N is 5, so you will have your 5 minus minus your R. Now, for A, it is the first term. In this case, it will be your 1. Then, you have your B, which will be this one. In this case, it will be your X to the power of R like that.

[00:01:15]So, that's that. Now, why is it T R + 1?

[00:01:19]So, this R + 1 tells you the term. Like, for example, if you're looking at the term independent of X, you're looking at the term. For example, if you have if you're expanding, let's say, the binomial expansion X + 1 like that to the power 6, you will realize to say the first term will be N combination of 0.

[00:01:41]Then, you have, let's say, you have your your 1 to the power 5 X to the power 0 like that. Plus, you have your N combination 1 * your 1 to the power 4.

[00:01:52]Then, you have your X to the power 1.

[00:01:55]It's continuous. So, you can see that this one is the first term and its error is zero.

[00:02:02]This is the second term and its error is one. So, second term, first term, error is equal to zero. Second term, error is equal to one. So, from this you can establish that error is less than n by one. So, n minus one. So, not necessarily n minus one, the position. So, the term minus one, that's will be equal to your error, like that. So, this is why we say the term you are looking for is error plus one.

[00:02:38]That means to say if it is the first term, you will say that your error is one zero.

[00:02:45]Your error when it's zero plus one, that will be your term, which will be equal to term number one. So, term number one will have an error of zero. Term number two will have an error of one. So, it will be one plus one, which will be term number two, like that. So, that's the formula for that one. So, with that said, let's deal with this. So, if we have our x minus one over x to the power of eight, if I'm not mistaken. So, this one, using the formula term t r plus one should be equal to we have our our n combination r, a n minus r b r. We are going to replace. We know that our n in this case is eight. So, we are going to have our eight combination r, then our a is equal to x in this case. So, the first term is your a. Then to the power of your eight, n is equal to eight minus r, like that. Then your b, which is equal to -1 over X to the power of R, which you don't know.

[00:03:57]Now, term like that.

[00:04:00]Now, one thing is he wants to eliminate X.

[00:04:04]So, what do you do? First, you split this. You're going to have your eight combination of R, then you have your X 8 - R, then this R can be distributed as -1 to the power of R, then you have times your X. If X goes up, it will take the negative R. Or if you want, you can say negative one like that, then we have our R like that. So, what do we have? We have our eight combination of R, then we have our X to the power of 8 - R, then times we have our -1 to the power of R like that.

[00:04:46]So, then we have our -1 times R, it will be equal to X to the power negative R like that. So, all we want is this X to disappear. So, this is our term that we are interested in.

[00:05:03]And this term. So, since that's the term we are interested in, we are going to isolate them and multiply them and say the only way we can eliminate them. We are already multiplying them. So, we multiply them. So, we are going to say if we multiply this times the X to the power negative R. So, we want a term that doesn't have X. So, we are going to equate it to X to the power zero.

[00:05:25]Because why? We know that X to the power zero is equal to one. So, that's why we equate it to X to the power zero so that we can find the value of R. So, we are going to say this times this, we keep one base at the power, so it will be 8 - R. We have this - R should be equal to X to the power zero. We cancel this, we cancel this we are going to have eight minus negative r minus r i to be negative two r should be equal to zero we take this value here we are going to have eight is equal to two r we divide two we divide two so our r value is equal to four so now your t was saying r plus one so that means to say that term will be equal to the term will be equal to four plus one which is equal to term number five if your r is four your term is number five so now let's find the value we now know the value of r for the equation that says x minus one over x to the power of eight so we are going to have something that is like this so t r plus one which we have established that it is term number five should be equal to n combination of r then we have our a n minus r b r so replacing we have n is this value which is our eight combination of r which is our four then times our a a which is a our a which is equal to we have this x so we are going to say x n is 10 or n is eight minus our r which is four then our b is equal to our negative one over our x like that to the power of our r which is our four like that so we are going to find that it will be eight eight combination four so this eight combination four is the same as eight factorial over eight minus four eight minus four factorial then repeat the four factorial like that times your X 8 - 4 is 4 times uh -1 times uh 4 1 to the power -4 1 -1 to the power 4 it will be equal to positive 1. Any number to an even number as long as it's in brackets it's always positive. Then we have over our value which will be equal to X to the power of 4. So this and this cancel. So if they cancel that's all that you're doing the correct thing. So we have uh 8 factorial over we're remaining with 4 factorial times 4 factorial. So what we're going to do is we're going to say 8 factorial is the same as 8 times 7 times 6 times 5 times then 4 factorial like that over the 4 factorial times 4 factorial. Why have we ended on 4 factorial? 4 factorial is the same as 4 times 3 times 2 times 1 which by leaving it like this we haven't changed anything. So that we can cancel cancel.

[00:08:50]Then we are remaining with 8 times 7 times 6 times 5 over 4 factorial is the same as 4 times 3 times 2 times 1. So that this cancels with that we're going to get a 2. This cancels with this. This cancels with this.

[00:09:11]Or we can yeah, this cancels with this to get a 2. So what we are remaining with is just a 7 times 2 times 3 over 1.

[00:09:21]So the answer here it will be equal to Yeah, so this is 5 not 3.

[00:09:29]So 8 times 7 times 6 times 5 So this is going to give us two times uh Okay, two times five, that's 10. 10 times seven, that's 70. So, uh that is the term independent of X, so it is the term number five like that.

[00:09:51]Uh let's look at another example. Let's say we have uh 3 X minus 2 over X squared like that to the power of 18.

[00:10:01]So, same approach for this one. We are going to say uh T R 1 R plus 1 should be equal to N combination 1.

[00:10:12]N combination R, sorry. A N minus R B R.

[00:10:17]So, your A is this one. So, we are going to say your N is the power which is your 18.

[00:10:24]Combination of your R you don't know.

[00:10:27]Then you have your A which is the 3 X in brackets.

[00:10:32]Then our N will be equal to 18 minus R. Then our B will be equal to the negative of 2 over our X squared.

[00:10:46]Then the power is R like that.

[00:10:49]So, we have 18 combination R.

[00:10:54]We can split this by saying this is the same as 3 to the power of 18 minus R.

[00:11:01]Then we have our X to the power of 18 minus R. So, it can be distributed like that.

[00:11:11]Then we have our B which is equal to negative 2.

[00:11:18]We have negative two like that.

[00:11:23]We have our negative two. It can also be split by raising it to the power R times this one it goes up it will be X to the power negative two then the overall power is R. So it has gone up that's why it's negative two. So we are going to have 18 combination R then we have three 18 minus R X 18 minus R then we have times our negative two in brackets to the power of R then we have negative two times R it will be equal to a X to the power negative two R. So remember the same approach where that's we are just interested in this. So we are going to say X to the power 18 minus R times X to the power negative two R should be equal to X to the power of zero. So this is going to give us a value that is equivalent to we we take one base at the power so it will be 18 minus R then we have minus two R should be equal to X to the power zero like that.

[00:12:37]So this and this one cancel then this is going to give us 18 minus R minus two R should be equal to zero. So we have 18 negative R minus R it will be negative three R will be equal to zero.

[00:12:53]So 18 will be equal to we take this this side it will be three R we divide our R value should be equal to six like that.

[00:13:03]So that means to say T R plus one is equal to T six plus one which will be equal to the seventh term which makes sense. The seventh term will have an R of um The seventh term will be have an R of six. So to find the value now we have the value for three X minus two over X squared to the power of 18. We know that the formula is saying that we have n combination r a n minus r b r. So, in this case, our n will be equal to 18 combination of r is 6. Then, we have our a is 3x. So, we're going to have 3x to the power of our r our n is 18 minus our r is 6 in this case. Then, our b is equal to -2 over our x squared to the power of our r is 6.

[00:14:05]So, this is what we're going to do.

[00:14:07]Then, our 18 combination r is the same as 18 over 18 minus 6 factorial. Then, we have our 6 factorial repeated like that. Times, we have our three to the power of a That's three to the power of Okay, 3x.

[00:14:29]Then, we have 6 18 minus 6, that's 12.

[00:14:33]Then, we have times our -2 to the power of our 6 over our x to the power of 2 * 6 like that.

[00:14:47]So, what are we going to end up with?

[00:14:50]So, we have our 18 factorial. So, we're going to say 18 factorial over our 18 minus 6, that's 12 factorial times our 6 factorial. Then, we have times the three to the power of 12.

[00:15:08]We also have our x to the power of 12.

[00:15:10]So, it can be distributed like that.

[00:15:13]Then, we have times -2 to the power 6, that will be equal to a 64 like that.

[00:15:21]Because two 2 times two, that's four. So, two times two times two times two times two times two.

[00:15:30]Two times two, four times two, eight times two, 16 times two, that's two times two, 64. So, that's how come we have a 64 there. So, this one now, 18 will be 18 times uh 18 times 17 times 16 times uh 15 times 14 times 13 times uh 12. So, 12 factorial because we have a 12 there. Let's Let's just move this a bit.

[00:16:03]So, if we move it here, we have 12 factorial there. Then, we have times three to the power of 12. X to the power 12.

[00:16:12]What are we forgetting down here?

[00:16:14]Two times uh two times six, that will be 12 like that.

[00:16:22]So, we have this value times uh 64 over uh X to the power 12. So, we cancel this, we cancel this.

[00:16:37]We have uh over 12 factorial times six factorial.

[00:16:42]So, to reduce, we're going to end up with this and this cancels. Then, we have uh 18 times 17 times uh 16 times 15 times 14 times 13 over six is the same as six times five times uh four times three times uh two times one.

[00:17:09]Then, we have times uh three to the power 12. We are not going to do anything with this value. Times uh 64.

[00:17:16]So, all we can do here is uh reduce some values of which when it reduce so five goes we are remaining with three. Four goes we are going to remain with uh four divided by 16 that's four.

[00:17:31]If we remove our six goes we are going to remain with three here. If we remove our two it will be seven. If we remove our three let's just cancel this. So, we are going to remain with our three times 17 times our four times seven times our 13 like that.

[00:17:59]Then we have times our three to the power 12 times our 64 Okay, so this is our value. We can just multiply some values uh like the four 17 times seven times 13. Okay, let's begin with our biggest values here. We can have our 13 17 times our 13 which will give us this times this it will give us 21 51 This times this it will be seven one.

[00:18:37]So, this will be one two remainder one 221. So, we have this and this is 221.

[00:18:45]So, we write our 221. Let's multiply 221 by seven. This will give us a value of seven four remainder one 15. So, we have 1,547 So, we have multiplied this this this.

[00:19:06]Let's just multiply by our four.

[00:19:09]If we multiply four we are going to get a 7 by 4, that's 28.

[00:19:15]4 by 4, that's 16 plus 2 remain the two remaining from 28, it will be our eight. This times this is 20 plus the one remaining, it will be 21, so one remainder two. 4 by this it will be 4 plus 1 plus two, sorry. It will be 6.

[00:19:38]So, this is the value we are getting.

[00:19:42]6,188 Now, we are just remaining with We have multiplied this this this this. So, let's just multiply with three. So, we have this value times uh three.

[00:19:56]This is going to give us 8 times 3, it will be 24.

[00:20:01]Remainder two, so 24 again. 24 plus two, that's six. This times this, it will be 3 plus that's remainder two, it will be five. This times this is 18. So, we are just going to expand it as far as that to give us our value which we have said it's to be 18 uh like this.

[00:20:27]Times uh 3 to the power 12 times uh 64 like that. So, now this this formula is not only used to find the term independent, but you can also use it to find any given term.

[00:20:42]Like for example, let's find the third term in this expansion.

[00:20:46]So, let's say we are required to find the third term. So, if you're looking at third term, term number three, r will be equal to two. So, the r will be less than the term itself by one. So, we are going to have the formula t r plus one should be equal to we have our n combination r a n minus r b r like that. So, our n in this case is equal to three combination of r is two. Then we have our a which is uh our 2x in this case. Then we have our three minus our one like that then times our b which is equal to negative y to the power of r which is equal to two.

[00:21:33]So, we have This is the same as three factorial over three minus two factorial then two factorial times we have our 2x to the power of three minus one that's two. Then we have times negative y squared it will be positive y squared like that.

[00:21:53]So, we are going to have our three factorial over three minus two that's uh that's one factorial two factorial times two squared that's four. X squared that's x squared. Then we have times y to the power of two like that.

[00:22:13]So, with that said we are going to have um our value which will be equal to three is the same as three times two times one over we have our two two factorial is just two. So, this and this cancels. Then we have times Okay, so if you look here we are supposed to write minus two r is equal to two.

[00:22:37]So, minus two minus two two which will give us one here.

[00:22:45]So, three if we say three minus two it's one. Then we have our 2x not 2x squared.

[00:22:53]Yeah. So, everything is okay.

[00:22:56]Then what we have is times 2x then we have y squared because this times this it will be 2 x y squared. So, we have 3 * 2 x y squared, which will give us 3 * 2 it will be 6 x y squared like that. So, this is how we solve that one. Thank you very much for watching. Hope you would like, subscribe, and share for more to avoid missing out this very powerful content. See you in the next tutorial.