MHB Find Term with $x^2$ in Binomial Theorem

[paulmdrdo1]

find the term with $x^2$

$\displaystyle\left(x^2-\frac{1}{x}\right)^{10}$

thanks!

 

[paulmdrdo1]

Re: binomial theorem

i did some some work here using this formula $\displaystyle a^{n-r}b^r$

 

[MarkFL]

Re: binomial theorem

According to the binomial theorem, what will the general term look like?

 

[paulmdrdo1]

Re: binomial theorem

the literal parts must be of the form $a^{n-r}b^r$ and the coefficient can be determine by using this formula $\frac{n(n-1)(n-2)...(n-r+1)}{r!}$ but i don't know how to apply this to the problem because it is a binomial in terms of x only.

 

[MarkFL]

Re: binomial theorem

The binomial theorem states:

$$(a+b)^n=\sum_{k=0}^{n}\left({n \choose k}a^{n-k}b^{k} \right)$$

What are $a$ and $b$ in this problem?

 

[paulmdrdo1]

Re: binomial theorem

$\displaystyle a=x^2$ $\displaystyle b=\frac{1}{x}$

 

[MarkFL]

Re: binomial theorem

$\displaystyle a=x^2$ $\displaystyle b=\frac{1}{x}$

Good, so what is the general term then, based on the binomial theorem?

 

[paulmdrdo1]

Re: binomial theorem

oh no. i have no idea how to do that all i know is the term being asked is the 9th term.

because n-r= 2--->10-r=2 ---->r=8. ith term = 8+1=9th term.

 

[MarkFL]

Re: binomial theorem

That is similar to what I have in mind. The general term is:

$${10 \choose k}\left(x^2 \right)^{10-k}\left(\frac{1}{x} \right)^k$$

This can be simplified as follows:

$${10 \choose k}x^{2(10-k)}\cdot x^{-k}={10 \choose k}x^{20-3k}$$

Now, we want the exponent on $x$ to be $2$, so what must the value of $k$ be?

 

[paulmdrdo1]

Re: binomial theorem

k should be 6. but still I'm kind of confused of what is that "k" stand for.

 

[MarkFL]

Re: binomial theorem

k should be 6. but still I'm kind of confused of what is that "k" stand for.

$k$ is the index of summation in the binomial theorem as I gave it above. Since $n=10$, there will be $11$ terms, for $k=0$ to $k=10$. $k=6$ represents the $7$th term.

So, with $k=6$, what is this $7$th term?

 

[paulmdrdo1]

Re: binomial theorem

7th term is 210x^2

 

[MarkFL]

Re: binomial theorem

7th term is 210x^2

Yes, that's correct. :D