Question

Find the value of the missing coefficient in the factored form of 8f^3 -216g^3
.

8f^3 -215g^3 = (2f-6g)(4f^4 +?fg+36g^3)
The value of
blank.

The solution is

Answer 1

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.

1. **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 \( b = 6g \):
\[
(2f - 6g)((2f)^2 + (2f)(6g) + (6g)^2) = (2f - 6g)(4f^2 + 12fg + 36g^2)
\]

2. **Confirm the factored form**:
The factorization yields:
\[
8f^3 - 216g^3 = (2f - 6g)(4f^2 + 12fg + 36g^2)
\]

3. **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:
\[
\boxed{12}
\]