To determine which worked equation correctly demonstrates polynomial division by recognizing it as the inverse operation of multiplication, we need to evaluate how to divide the polynomial [tex]\(8x^2 - 4x + 12\)[/tex] by [tex]\(4x\)[/tex]:
1. First, let's write down the polynomial division:
[tex]\[
\frac{8x^2 - 4x + 12}{4x}
\][/tex]
2. To divide each term of the polynomial [tex]\(8x^2 - 4x + 12\)[/tex] by [tex]\(4x\)[/tex], we should do the following:
[tex]\[
\frac{8x^2}{4x} - \frac{4x}{4x} + \frac{12}{4x}
\][/tex]
3. Simplify each term:
[tex]\[
\frac{8x^2}{4x} = 2x, \quad \frac{4x}{4x} = 1, \quad \text{and} \quad \frac{12}{4x} = \frac{3}{x}
\][/tex]
4. Combining these results, we have:
[tex]\[
2x - 1 + \frac{3}{x}
\][/tex]
Now, let’s examine each option to identify which one accurately demonstrates this operation.
1. [tex]\( \frac{8x^2 - 4x + 12}{4x} = (4x)(8x^2 - 4x + 12) \)[/tex]
This equation implies multiplication, not division. Therefore, it is incorrect.
2. [tex]\( \frac{8x^2 - 4x + 12}{4x} = (-4x)(8x^2 - 4x + 12) \)[/tex]
Similarly, this implies multiplication with a negative factor, which is also incorrect.
3. [tex]\( \frac{8x^2 - 4x + 12}{4x} = \left(-\frac{1}{4x}\right)\left(8x^2 - 4x + 12\right) \)[/tex]
This does not correctly reflect the division process as demonstrated by simplifying above.
4. [tex]\( \frac{8x^3 - 4x + 12}{4x} = \left(\frac{1}{4x}\right)\left(8x^2 - 4x + 12\right) \)[/tex]
This option attempts to represent a division, but the left side equivalent contains [tex]\(8x^3\)[/tex] instead of [tex]\(8x^2\)[/tex].
After examining these eliminations, it turns out that the correct representation of polynomial division in the given choices is not perfectly displayed. However, following the methods and steps used for division, the closest manipulated polynomial seems to have intended the correct idea.
Thus, the selected answer:
[tex]\[
\boxed{4}
\][/tex]
