SOLUTION: Find the coefficient of b^16 in the binomial expansion of (2b^3- 1/(4b^2))^12

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Question 1043501: Find the coefficient of b^16 in the binomial expansion of
(2b^3- 1/(4b^2))^12
Found 3 solutions by Boreal, Edwin McCravy, MathTherapy:
Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
I am writing this as (2b^3-(1/4)b^2)
The first term will never be b^16, only the second term, and that is when it is raised to the 8th power.
That occurs at the binomial expansion 12C8 (2b^3)^4-(4b^2)^8=495{16b^12+(1/65536)b^16).

Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!
%282b%5E3-+1%2F%284b%5E2%29%29%5E12

The (R-1)st term is

%2812CR%29%282b%5E3%29%5ER%2A%281%2F%284b%5E2%29%29%5E%2812-R%29

Remove the parentheses in the second factor by
multiplying exponents.  Write 4 as 22

%2812CR%292%5ER%2Ab%5E%283R%29%2A%28expr%281%2F%282%5E2%2Ab%5E2%29%29%29%5E%2812-R%29

Write the factors in the denominator out of the
denominator with negative exponents:

%2812CR%292%5ER%2Ab%5E%283R%29%2A%282%5E%28-2%29%2Ab%5E%28-2%29%29%5E%2812-R%29

Remove the parentheses by multiplying exponents:

%2812CR%292%5ER%2Ab%5E%283R%29%2A2%5E%28-24%2B2R%29%2Ab%5E%28-24%2BR%29

Add the exponents of 2 and add the exponents of b:

%2812CR%292%5E%28R-24%2B2R%29%2Ab%5E%283R-24%2BR%29%29

Combine terms in the exponents:

%2812CR%292%5E%283R-24%29%2Ab%5E%284R-24%29%29

Since we want the term in b16, we
set the exponent of b equal to 16

4R-24 = 16
   4R = 40
    R = 10

%2812C10%292%5E%283%2A10-24%29%2Ab%5E%284%2A10-24%29%29

66%2A2%5E%2830-24%29%2Ab%5E%2840-24%29%29

66%2A2%5E6%2Ab%5E16%29

66%2A64%2Ab%5E16%29

4224b%5E16%29

Coefficient of b16 = 4224

Edwin

Answer by MathTherapy(10557) About Me  (Show Source):
You can put this solution on YOUR website!

Find the coefficient of b^16 in the binomial expansion of
(2b^3- 1/(4b^2))^12
The 5th term of the expansion: %282b%5E3+-+1%2F%284b%5E2%29%29%5E12 contains: b%5E16, and has a coefficient of: highlight_green%28495%29.