To determine which equation correctly demonstrates that division of polynomials can be recognized as the inverse operation of multiplication, we need to look for an expression that properly shows the relationship between division and multiplication.
The original expression is:
\[ \frac{8x^2 - 4x + 12}{4x} \]
When we think about division as the inverse of multiplication, we can express it in the form:
\[ \frac{A}{B} = C \implies A = B \cdot C \]
For example, if we let:
\[ A = 8x^2 - 4x + 12 \] \[ B = 4x \] \[ C = \text{some multiplier} \]
We want to find a multiplier such that:
\[ A = B \cdot C \Rightarrow 8x^2 - 4x + 12 = 4x \cdot C \]
Thus, \( C \) would be equal to \(\frac{8x^2 - 4x + 12}{4x}\).
Next, let's analyze the response options:
- \(\frac{8x^2 - 4x + 12}{4x} = (-4x)(8x^2 - 4x + 12)\) - Incorrect.
- \(\frac{8x^2 - 4x + 12}{4x} = (-\frac{1}{4x})(8x^2 - 4x + 12)\) - Incorrect.
- \(\frac{8x^2 - 4x + 12}{4x} = (14x)(8x^2 - 4x + 12)\) - Incorrect.
- \(\frac{8x^2 - 4x + 12}{4x} = (\frac{1}{4x})(8x^2 - 4x + 12)\) - Correct.
- \(\frac{8x^2 - 4x + 12}{4x} = (4x)(8x^2 - 4x + 12)\) - Incorrect.
Among all responses, response 4 is the correct answer.
\[ \frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right)(8x^2 - 4x + 12) \]
This shows that dividing \(8x^2 - 4x + 12\) by \(4x\) is the same as multiplying \(8x^2 - 4x + 12\) by \(\frac{1}{4x}\), thus demonstrating that division can be seen as the inverse operation of multiplication.