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Find the coefficient of u^2 v^9 in the expansion of (2u^2 - v^3)^4.

 Jul 5, 2022
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Hi Guest!

\(\text{The general term is:}\)

  \(4Ck (2u^2)^{4-k}(-v^3)^k\)  \(\text {(By the binomial theorem)}\)

\(\text{We want the coefficient of:}\)  \(u^2v^9\)

 

\(\text{So, we want}\):  \((-v^3)^k=av^9 \implies k=3\)       

  (\(\text{Alternatively,}\) \((2u^2)^{4-k} =au^2 \implies {4-k}=1 \iff k=3\)\(\text{(where a is any constant to be determined, but the main point is to get the correct exponent).}\)

 

\(\text{Hence,}\)

\(4C3(2u^2)^{1}(-v^3)^3 = 4(2)u^2(-1)v^9=-8u^2v^9\)

\(\text{Therefore, the desired coefficient is: -8}\) 

I hope this help!

 Jul 5, 2022

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