Question
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=(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=(−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=(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=(−4x)(8x2−4x+12)
Answers
GPT-4o mini
To demonstrate the concept of dividing polynomials by recognizing division as the inverse operation of multiplication, we look for an equation where the division leads to a multiplication statement that is true.
From the provided options, we analyze each one:
1. **Option 1**:
\[
\frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right)(8x^2 - 4x + 12)
\]
This is true because multiplying \(\frac{1}{4x}\) by \(4x\) results in \(1\), thus demonstrating the inverse.
2. **Option 2**:
\[
\frac{8x^2 - 4x + 12}{4x} = \left(-\frac{1}{4x}\right)(8x^2 - 4x + 12)
\]
This is incorrect because it introduces a negative sign that is not justified by the equation.
3. **Option 3**:
\[
\frac{8x^2 - 4x + 12}{4x} = (4x)(8x^2 - 4x + 12)
\]
This is incorrect because it implies that multiplying \(4x\) by \(8x^2 - 4x + 12\) would yield \(8x^2 - 4x + 12\), which is not true.
4. **Option 4**:
\[
\frac{8x^2 - 4x + 12}{4x} = (-4x)(8x^2 - 4x + 12)
\]
This is also incorrect for the same reason as option 3.
Based on the analysis, **Option 1** correctly demonstrates the idea that division can be seen as an inverse operation of multiplication:
\[
\frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right)(8x^2 - 4x + 12)
\]
From the provided options, we analyze each one:
1. **Option 1**:
\[
\frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right)(8x^2 - 4x + 12)
\]
This is true because multiplying \(\frac{1}{4x}\) by \(4x\) results in \(1\), thus demonstrating the inverse.
2. **Option 2**:
\[
\frac{8x^2 - 4x + 12}{4x} = \left(-\frac{1}{4x}\right)(8x^2 - 4x + 12)
\]
This is incorrect because it introduces a negative sign that is not justified by the equation.
3. **Option 3**:
\[
\frac{8x^2 - 4x + 12}{4x} = (4x)(8x^2 - 4x + 12)
\]
This is incorrect because it implies that multiplying \(4x\) by \(8x^2 - 4x + 12\) would yield \(8x^2 - 4x + 12\), which is not true.
4. **Option 4**:
\[
\frac{8x^2 - 4x + 12}{4x} = (-4x)(8x^2 - 4x + 12)
\]
This is also incorrect for the same reason as option 3.
Based on the analysis, **Option 1** correctly demonstrates the idea that division can be seen as an inverse operation of multiplication:
\[
\frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right)(8x^2 - 4x + 12)
\]