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Which of the following worked equations demonstrates that you can divide polynomials by recognizing division as the inverse operation of multiplication?(1 point) Responses 8x2−4x+124x=(4x)(8x2−4x+12) Start Fraction 8 x squared minus 4 x plus 12 over 4 x End Fraction equals left parenthesis 4 x right parenthesis left parenthesis 8 x squared minus 4 x plus 12 right parenthesis 8x2−4x+124x=(−14x)(8x2−4x+12) Start Fraction 8 x squared minus 4 x plus 12 over 4 x End Fraction equals left parenthesis negative Start Fraction 1 over 4 x End Fraction right parenthesis left parenthesis 8 x squared minus 4 x plus 12 right parenthesis 8x2−4x+124x=(14x)(8x2−4x+12) Start Fraction 8 x squared minus 4 x plus 12 over 4 x End Fraction equals left parenthesis Start Fraction 1 over 4 x End Fraction right parenthesis left parenthesis 8 x squared minus 4 x plus 12 right parenthesis 8x2−4x+124x=(−4x)(8x2−4x+12)

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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:

  1. \[ \frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right)(8x^2 - 4x + 12) \]

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: \[ \frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right)(8x^2 - 4x + 12) \]

What does the fraction look like

When representing the expression \(\frac{8x^2 - 4x + 12}{4x}\) as a fraction, it looks like this:

\[ \frac{8x^2 - 4x + 12}{4x} \]

In this representation:

  • The numerator is \(8x^2 - 4x + 12\).
  • The denominator is \(4x\).

If you're summing it up in the context of showing the inverse relationship between multiplication and division, this fraction signifies that dividing the polynomial \(8x^2 - 4x + 12\) by \(4x\) can be expressed as multiplying \( (8x^2 - 4x + 12) \) by \(\frac{1}{4x}\):

\[ \frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right)(8x^2 - 4x + 12) \]

This shows that dividing by \(4x\) is the same as multiplying by \(\frac{1}{4x}\). The fraction is laid out clearly with the polynomial over the term.