SOLUTION: find the term independent of x in the expansion of {2x^3 - 1/4(x^5)}^8

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Question 1117568: find the term independent of x in the expansion of {2x^3 - 1/4(x^5)}^8
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


To get a term independent of x, you need to take the "2x^3" term from 5 of the factors and the "1/(4x^5)" term from the other 3 factors. So the term is

C%288%2C5%29%2A%282x%5E3%29%5E5%2A%281%2F%284x%5E5%29%29%5E3
%2856%29%2A%2832x%5E15%29%2A%281%2F%2864x%5E15%29%29
56%2A32%2F64
28

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tutor ikleyn has pointed out that I lost track of a sign in the above response....

The "1/4x^5" term in the binomial is negative, so the constant term is

C%288%2C5%29%2A%282x%5E3%29%5E5%2A%28-1%2F%284x%5E5%29%29%5E3
%2856%29%2A%2832x%5E15%29%2A%28-1%2F%2864x%5E15%29%29
-56%2A32%2F64
-28

Answer by ikleyn(52771) About Me  (Show Source):
You can put this solution on YOUR website!
.

            The correct answer is -28.

            See the full and correct solution below. 


The binomial expansion is this formula

 
%28a%2Bb%29%5En = a%5En + C%5Bn%5D%5E1%2Aa%5E%28n-1%29%2Ab + C%5Bn%5D%5E2%2Aa%5E%28n-2%29%2Ab%5E2 + C%5Bn%5D%5E3%2Aa%5E%28n-3%29%2Ab%5E3 + . . . + C%5Bn%5D%5E%28n-1%29%2Aa%5E1%2Ab%5E%28n-1%29 + b%5En


In our case,  n = 8,  a = 2x%5E3,  b = -1%2F4x%5E5,  therefore, the binomial expansion in our case is  


%282x%5E3-+1%2F%284x%5E5%29%29%5E8 = %282x3%29%5E8 + C%5B8%5D%5E1%2A%282x%5E3%29%5E7%2A%28-1%2F%284x%5E5%29%29 + C%5B8%5D%5E2%2A%282x%5E3%29%5E6%2A%28-1%2F%284x%5E5%29%29%5E2 + C%5B8%5D%5E3%2A%282x%5E3%29%5E5%2A%28-1%2F%284x%5E5%29%29%5E3 + . . . + C%5B8%5D%5E9%2A%282x%5E3%29%2A%28-1%2F%284x%5E5%29%29%5E7 + %28-1%2F%284x%5E5%29%29%5E8.


The general term of this expansion is  C%5B8%5D%5Ek%2A%282x%5E3%29%5E%288-k%29%2A%28-1%2F%284x%5E5%29%29%5Ek =  = C%5B8%5D%5Ek%2A%28-1%29%5Ek%2A2%5E%288-k-2k%29%2Ax%5E%283%2A%288-k%29-5k%29.


The term independent of x is at  3*(8-k) - 5k = 0,  or


    24 - 3k - 5k = 0  ====>  24 = 8k  ====>  k = 3   (the fourth term).


Then the coefficient at this term is  C%5B8%5D%5E3%2A2%5E%288-3-2%2A3%29%2A%28-1%29 = %28%288%2A7%2A6%29%2F%281%2A2%2A3%29%29%2A%28-1%29%2A2%5E%28-1%29 = -%288%2A7%29%2F2 = -28.

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The solution by @greenestamps giving the answer  "28"  contains an error.