6 Binomial Theorem - Example 2 - Finding The Term Independent Of X.
Now, let's take a look at example to find the term, independent of X in the expansion of X square, minus 2 of X raised to the power 12. Now, the term independent of X means the constant term, the constant term, whereby the X power is 0. Okay. So it's actually talking about X to the power of 0. Alright. So how are we going to find the specific term with X to the power 0? All right?
Of course, we got to use the general term formula. All right. So the first step will be, of course, let the term that we are.
Interested in be the general term. So the general term according to what we just talked about will be 12, CR, okay, X square. And of course, the second term negative 2 over X right now, negative 2 of X will be having a power of our well X square. We have a power of 12 minus R. Okay. So this is the general term in this binomial expansion. What it means is that where every single term inside this expansion should look like this. Okay, we have a blueprint of something like this.
Alright. So depending on. What is the value for R it's, something the values for R, you will get each individual terms. Alright? So, for example, if you want when you substitute in R equals to 3, okay, you will get the fourth time. We substitute in R equals to 5. You get a sixth term and so on and so forth.
But the problem is we have no idea where is this constant term? Where is this term this independent of X? I mean, is it in the 15 is it? The sixth term? Well, we have no idea and that's why we don't know what is our R? Okay?
So now the. First thing we have to do would be to find out what is the value of R, okay, and how are we going to do that? Well, basically, we don't know anything yet. The only thing we know is that the term that we want to find we get the term that we want to find has got an X to the power of 0. Okay. So we have the menus of that.
And how do we mean of that we make use of the X that we have, so we'll, just draw all the XS that we have. So this is x squared raised to the power of 12 minus R. Okay. And of course, and this X. Here is actually X to the power of negative.
One okay, raised to the power of r all right is equal to X to the power of 0 all right now. Take notice of how I ignore all the constants. Ok, 12 see R is a constant.
They have no idea why this. And therefore it is totally useless for me at the moment. Ok? Negative 2 again is another constant again, I don't know why R? And therefore it is totally useless for me. So the only thing that we take note of is 2x, okay, so we can compare the power from with the left hand. Side, okay, with the right-hand side.
Okay. So the power of the left-hand side with the right size. So let us continue now using a law of indices, we know that we can expand the powers here. So we have actually X to the power of 24, minus 2, R multiplied by X to the power of negative R is equal to X to the power of 0. So from here, okay, by comparing the power, we can say that, well, 24 minus 3 R is equal to 0. Okay.
And therefore from here, we know that our R is equal to 8 now. What do we know that when R? Is equal to 8, we will have X to the power of 0 is that simple that is it. This is what is going on here. Ok?
So when R is equal to 8, we have X to the power of 0. And now what to do? What are we supposed to do? Well, all we have to do now is to substitute in R equals 2 into this term. Remember that this is the term that we're interesting.
So all we have to do is substitute in our. It goes to 8, ok? And we will get 12. See 8, ok. Now, X square, 12, minus 8 that will give us a form right? And of course, negative 2 X. Raised to the power of 8 okay.
So from your calculator, you will know that 12 see 8 is 495. Okay. And then, of course, this term here we have X to the power of 8 and negative 2 raised to the power of 8 that give us 2 5, 6, ok, positive and the denominator by X to the power 8 so from here, except I like, except right here can cancel. And the final answer will have well, 120 6700 M twin D. So this is the term that we're interested in the term that is independent of X.
The term let's go X to the power of 0. The. Constant term, I mean, whatever we want to know it. Okay, they all mean, the same thing X to the power of 0. Now, if the question asked you to find say, you know, for example, the term in this expansion with X cube. Okay. Now, let's say that they're no longer interested in the term, independent of X, not interested in the term.
We execute more how we can do that. Ok. Now, exactly the same. Okay? The steps here are exactly the same. In fact, you can even copy everything here and recycle this. Okay. It will be 24 minus 3.
R equals to 3, ok. So from here we work out, we know that 3 R is equal to 21, and therefore R is equal to 7. Now what this tells us is that when R is equal to 7, we will get the term that has got X cube. Ok. Now, bear in mind that your R value.
Okay, whatever answer they get for your R. In this case we get it. Or in this case, we get 7. Your R has to be positive, has to be an integer. Okay, you cannot have said R equals to your know, 3 over 2.
Now, this is definitely wrong all right. All right? You can't say that where. You can get R equals to negative 1 again.
This is certainly wrong right? So when this happens, okay, when you do get R in funny numbers like this, oh, you certainly have done something seriously wrong up here. So you better check, ok, go through your answers. One more time to see where you went wrong. We have now.