Binomial Expansion: Coeff of x^29 in (1+x^5+x^7+x^9)

In summary, the coefficient of x^{29} in the expansion of (1+x^{5}+x^{7}+x^{9})^{16} is zero. This is because there is no x^{29} term in the expansion, and all combinations of powers that could potentially give 29 result in a coefficient of 0.
  • #1
icystrike
445
1

Homework Statement


Find the coefficient of [tex]x^{29}[/tex] in the expansion of [tex](1+x^{5}+x^{7}+x^{9})[/tex].

Homework Equations


The Attempt at a Solution

 
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  • #2
It is zero: there is no [itex]x^{29}[/itex] term in
[tex]
(1+x^{5}+x^{7}+x^{9})
[/tex]

I suppose you are missing some power?
 
  • #3
CompuChip said:
It is zero: there is no [itex]x^{29}[/itex] term in
[tex]
(1+x^{5}+x^{7}+x^{9})
[/tex]

I suppose you are missing some power?

oh yea sorry.
its
[tex]
(1+x^{5}+x^{7}+x^{9})^{16}
[/tex]
 
  • #4
So you have 16 factors of
[tex]
(1+x^{5}+x^{7}+x^{9})
[/tex]
multiplying. The first thing I'd do is check which unique combinations of powers give 29. For example, suppose that you open up the brackets. You will encounter terms with four factors of x^5, a factor of x^9 and all 1's otherwise, which gives x^29. Are there any other combinations?

Then, go through them one by one... you are looking for something of the form
[tex]x^9 \cdot x^5 \cdot x^5 \cdot x^5 \cdot x^5 \cdot 1 \cdot 1 \ cdot 1 \cdots[/tex]
(16 in total). How many different orders are there? I.e., when you again think about multiplying out the brackets, how many terms are there that give this expression?
 
  • #5
I tried using another method:

attachment.php?attachmentid=22445&stc=1&d=1260634010.jpg
 

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  • #6
Yeah, well, that was essentially what I was thinking of too.

And your answer is correct.
 
  • #7
thanks compuchip (=
 

Related to Binomial Expansion: Coeff of x^29 in (1+x^5+x^7+x^9)

1. What is the formula for binomial expansion?

The formula for binomial expansion is (a + b)^n = ∑(nCr)a^(n-r)b^r, where a and b are constants, n is the power, and nCr is the combination formula for choosing r objects from a set of n objects.

2. How do you find the coefficient of a specific term in a binomial expansion?

To find the coefficient of a specific term, you can use the combination formula (nCr) and the exponent of the term. The coefficient of a term with the exponent k can be calculated as nCk, where n is the power and k is the exponent of the term.

3. How do you apply binomial expansion in real-life situations?

Binomial expansion can be applied in various real-life situations, such as in finance for calculating compound interest, in probability for calculating the chances of an event occurring multiple times, and in genetics for predicting the inheritance of traits.

4. What is the significance of the coefficient of a term in binomial expansion?

The coefficient of a term in binomial expansion represents the number of ways that term can be obtained from the expansion. It also indicates the relative importance or weight of that term in the overall expansion.

5. How can I use binomial expansion to expand a binomial to a certain power?

To expand a binomial to a certain power, you can use the formula (a + b)^n = ∑(nCr)a^(n-r)b^r, where a and b are the constants in the binomial and n is the desired power. Then, you can use the combination formula (nCr) to calculate the coefficients of each term in the expansion.

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