To find the missing coefficient in the factored form of \( 8f^3 - 216g^3 \) given that it can be factored as \( (2f - 6g)(4f^2 + ?fg + 36g^2) \), we'll start by establishing the factorization and then comparing coefficients.
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Factor the difference of cubes: The expression \( 8f^3 - 216g^3 \) can be expressed as: \[ (2f)^3 - (6g)^3 \]
The difference of cubes can be factored as: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Here, set \( a = 2f \) and \( b = 6g \): \[ (2f - 6g)((2f)^2 + (2f)(6g) + (6g)^2) = (2f - 6g)(4f^2 + 12fg + 36g^2) \]
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Confirm the factored form: The factorization yields: \[ 8f^3 - 216g^3 = (2f - 6g)(4f^2 + 12fg + 36g^2) \]
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Identify the missing coefficient: From the comparison of the forms provided: \[ (2f - 6g)(4f^2 + ?fg + 36g^2) \] We see \( ?fg \) corresponds to \( 12fg \). Thus: \[ ? = 12 \]
Therefore, the value of the missing coefficient is: \[ \boxed{12} \]
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