Find the middle terms of this binomial expansion

In summary, the conversation discusses the expansion of ##\left( 4 - x^3 \right)^7## and the question asks for the middle terms of the expansion. It is unclear if the expansion has a middle term, but it is suggested that the 4th and 5th terms may be considered as the middle terms.
  • #1
songoku
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Homework Statement
Find the middle terms of ##\left( 4 - x^3 \right)^7##
Relevant Equations
Binomial expansion
I know how to expand it. My question is: the expansion has 8 terms so what would be the middle term? Will the answer be "the expansion has no middle term"?

Or maybe seeing the phrase of the question (middle terms), there will be two answers (the 4th and 5th term)?

Thanks
 
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  • #2
I assume the problem suggests all the expansion terms except obvious ##4^7## and ##-x^{21}##.
 
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  • #3
Thank you very much anuttarasammyak
 
  • #4
songoku said:
Homework Statement:: Find the middle terms of ##\left( 4 - x^3 \right)^7##
Relevant Equations:: Binomial expansion

I know how to expand it. My question is: the expansion has 8 terms so what would be the middle term? Will the answer be "the expansion has no middle term"?

Or maybe seeing the phrase of the question (middle terms), there will be two answers (the 4th and 5th term)?

Thanks
I suspect the intent of the questioner is to ask for the 4th and 5th terms as you suggested. However, the wording is not all that clear so @anuttarasammyak may indeed be correct.
 
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  • #5
Thank you very much SammyS
 

1. What is a binomial expansion?

A binomial expansion is a mathematical process that involves expanding a binomial expression, which is an expression with two terms, to a certain power. This process results in a polynomial with multiple terms. For example, (a + b)^2 = a^2 + 2ab + b^2.

2. How do you find the middle terms of a binomial expansion?

To find the middle terms of a binomial expansion, you can use the binomial theorem or Pascal's triangle. The binomial theorem is a formula that allows you to calculate the coefficients of each term in the expansion. Pascal's triangle is a visual representation that helps you determine the coefficients.

3. What is the purpose of finding the middle terms of a binomial expansion?

The middle terms of a binomial expansion can help you determine the coefficients and terms in the expanded polynomial. This can be useful in solving problems in algebra, calculus, and other areas of mathematics.

4. Can you provide an example of finding the middle terms of a binomial expansion?

Sure, let's say we want to find the middle terms of (x + y)^4. Using the binomial theorem, we can calculate the coefficients as follows: (4 choose 2) * x^2 * y^2 = 6x^2y^2. Therefore, the middle terms of the expansion are 6x^2y^2.

5. Are there any shortcuts or tricks for finding the middle terms of a binomial expansion?

Yes, there are a few shortcuts and tricks that can help you find the middle terms of a binomial expansion more quickly. For example, if the exponent of the binomial is even, the middle terms will always have the same exponent. If the exponent is odd, the middle terms will have exponents that differ by 1. Additionally, you can use patterns in Pascal's triangle to determine the coefficients without having to use the binomial theorem.

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