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# coefficient

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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