To demonstrate that you can divide polynomials by recognizing division as the inverse operation of multiplication, you need an equation that shows that dividing a polynomial by a term is equivalent to finding a factor of that polynomial.
Here’s the polynomial division presented:
Given: \[ \frac{8x^2 - 4x + 12}{4x} \]
You're looking for an equation that shows that:
\[ \frac{8x^2 - 4x + 12}{4x} = k(8x^2 - 4x + 12) \]
where \( k \) is some constant.
From the provided responses, the equivalent form of the equation should be structured correctly to show this division. The best option is:
- \[ \frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right)(8x^2 - 4x + 12) \]
This choice correctly represents the division by expressing the polynomial as a multiplication of a constant and a polynomial, confirming the inverse relationship between division and multiplication.
So, the correct response is: \[ \frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right)(8x^2 - 4x + 12) \]