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Here is the question,

 

Find the constant term in the expansion of,

\(\Big(x^2+\frac{1}{x}\Big)^4\)

 

 

 

When I expand this equation I get,

\(\frac{\left(x^3+1\right)^4}{x^4}\)

I think this is the correct expantion but I also got another answer,

\(\frac{x^{12}+4x^9+6x^6+4x^3+1}{x^4}\)

I dont know what the constant term is and if there is not what should I put as the answer ?

 

Thx in advance.

 Aug 13, 2019
 #1
avatar+6046 
0

\((a+b)^n = \sum \limits_{k=0}^n \dbinom{n}{k}a^k b^{n-k}\)

 

\(\left(x^2 + \dfrac 1 4 \right)^4 = \sum \limits_{k=0}^4 \dbinom{4}{k}(x^2)^k \left(\dfrac 1 4\right)^{n-k}\\ \text{The constant term appears when $k=0$}\\ \dbinom{4}{0}\left(\dfrac 1 4\right)^4 = \dfrac{1}{256} \)

.
 Aug 14, 2019
 #2
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Thank you!!!

 Aug 14, 2019
 #3
avatar+7761 
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Your expansion is correct! But you don't have any term that is independent of x, so the constant term is 0.

 Aug 14, 2019
 #4
avatar+6046 
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This isn't correct.  The constant term is 1/256 as shown above.

 

WA to verify this

 

https://www.wolframalpha.com/input/?i=(x%5E2+%2B+1%2F4)%5E4

Rom  Aug 14, 2019
 #5
avatar+28288 
+2

Rom has expanded \((x^2+\frac{1}{4})^4\)  but the expression in the original question is \((x^2+\frac{1}{x})^4\)

 

Given that the question asks for the constant term, Rom's version might make more sense; however, it is different from the original, for which Max's answer is correct.

Alan  Aug 15, 2019

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