Find Term with $x^2$ in Binomial Theorem

In summary, the conversation discusses the binomial theorem and finding the general term for a binomial expression. The general term is determined using the formula {n \choose k}a^{n-k}b^{k}, with a=x^2 and b=1/x. The value of k represents the index of summation and can be used to find a specific term in the binomial expression.
  • #1
paulmdrdo1
385
0
find the term with $x^2$

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

thanks!
 
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  • #2
Re: binomial theorem

i did some some work here using this formula $\displaystyle a^{n-r}b^r$
 
  • #3
Re: binomial theorem

According to the binomial theorem, what will the general term look like?
 
  • #4
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.
 
  • #5
Re: binomial theorem

The binomial theorem states:

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

What are $a$ and $b$ in this problem?
 
  • #6
Re: binomial theorem

$\displaystyle a=x^2$ $\displaystyle b=\frac{1}{x}$
 
  • #7
Re: binomial theorem

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

Good, so what is the general term then, based on the binomial theorem?
 
  • #8
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.
 
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  • #9
Re: binomial theorem

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

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

This can be simplified as follows:

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

Now, we want the exponent on $x$ to be $2$, so what must the value of $k$ be?
 
  • #10
Re: binomial theorem

k should be 6. but still I'm kind of confused of what is that "k" stand for.
 
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  • #11
Re: binomial theorem

paulmdrdo said:
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?
 
  • #12
Re: binomial theorem

7th term is 210x^2
 
  • #13
Re: binomial theorem

paulmdrdo said:
7th term is 210x^2

Yes, that's correct. :D
 

FAQ: Find Term with $x^2$ in Binomial Theorem

What is the binomial theorem?

The binomial theorem is a mathematical formula that allows us to expand binomials raised to any positive integer power. It can be written as (a + b)^n = a^n + nC1a^(n-1)b + nC2a^(n-2)b^2 + ... + b^n, where n is a positive integer and nCx is the binomial coefficient.

How do I find the term with $x^2$ in the binomial theorem?

To find the term with $x^2$ in the binomial theorem, we need to look for the term that has x^2 as its exponent. This term will have the form nCx(a^(n-x))(b^x), where nCx is the binomial coefficient, a is the coefficient of the first term, and b is the coefficient of the second term.

Can the binomial theorem be used for binomials with negative exponents?

Yes, the binomial theorem can be used for binomials with negative exponents. We can rewrite the binomial with negative exponents as a fraction, and then use the binomial theorem to expand it. For example, (a + b)^(-3) = 1/(a + b)^3 = 1/a^3 + 3/a^2b + 3/ab^2 + 1/b^3.

What is the purpose of finding the term with $x^2$ in the binomial theorem?

Finding the term with $x^2$ in the binomial theorem allows us to determine the coefficient of the x^2 term in the expanded form. This can be useful in solving equations or simplifying expressions involving binomials raised to a power.

Can the binomial theorem be used for expressions with more than two terms?

No, the binomial theorem can only be used for binomials, which are expressions with two terms. For expressions with more than two terms, we can use the multinomial theorem, which is an extension of the binomial theorem.

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