To determine which equation demonstrates that you can divide polynomials by recognizing division as the inverse operation of multiplication, let's analyze each option:
1. [tex]\(\frac{8 x^2-4 x+12}{4 x}=\left(\frac{1}{4 x}\right)\left(8 x^2-4 x+12\right)\)[/tex]
Here, the expression [tex]\(\left(\frac{1}{4 x}\right)(8 x^2 - 4 x + 12)\)[/tex] suggests that we multiply the polynomial [tex]\(8 x^2 - 4 x + 12\)[/tex] by the inverse of [tex]\(4 x\)[/tex]. This is correct because multiplying by the reciprocal or inverse is the same as dividing by the original value.
2. [tex]\(\frac{8 x^2-4 x+12}{4 x}=(4 x)\left(8 x^2-4 x+12\right)\)[/tex]
This equation states that dividing by [tex]\(4 x\)[/tex] is the same as multiplying by [tex]\(4 x\)[/tex], which is incorrect, as division by a number is not the same as multiplication by that number.
3. [tex]\(\frac{8 x^2-4 x+12}{4 x}=\left(-\frac{1}{4 x}\right)\left(8 x^2-4 x+12\right)\)[/tex]
This equation suggests that dividing by [tex]\(4 x\)[/tex] is equivalent to multiplying by the negative inverse [tex]\(-\frac{1}{4 x}\)[/tex], which is not true. The correct inverse should be positive [tex]\(\frac{1}{4 x}\)[/tex], not negative.
4. [tex]\(\frac{8 x^2-4 x+12}{4 x}=(-4 x)\left(8 x^2-4 x+12\right)\)[/tex]
This equation suggests that dividing by [tex]\(4 x\)[/tex] is the same as multiplying by [tex]\(-4 x\)[/tex], which is incorrect for the same reason as explained in option 2.
Given this detailed analysis, the correct worked equation that demonstrates polynomial division by recognizing division as the inverse operation of multiplication is:
[tex]\[
\frac{8 x^2-4 x+12}{4 x}=\left(\frac{1}{4 x}\right)\left(8 x^2-4 x+12\right)
\][/tex]
