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=(−4x)(8x2−4x+12) Start Fraction 8 x squared minus 4 x plus 12 over 4 x End Fraction equals left parenthesis negative 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)
Answers
GPT-4o mini
To demonstrate the concept of dividing polynomials using the inverse relationship of multiplication, we want to rearrange the division problem into a multiplication equation.
The expression we are working with is:
\[
\frac{8x^2 - 4x + 12}{4x}
\]
We want to express it in the form \( A = (B)(C) \), where \( A \) is the original expression, \( B \) is the divisor, and \( C \) is the quotient.
None of the options you provided seem to match perfectly with a standard interpretation of the division as multiplication. However, the concept being illustrated is that you can express the division as a product where one factor is the reciprocal of the divisor.
So, if we consider:
\[
\frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right) (8x^2 - 4x + 12)
\]
This would correspond to a choice where:
\[
\frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right)(8x^2 - 4x + 12)
\]
Given your options, the one that correctly demonstrates this relationship is:
**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**
Which can be simplified further if needed to express how dividing by \( 4x \) can be thought of as multiplying by its reciprocal.
The expression we are working with is:
\[
\frac{8x^2 - 4x + 12}{4x}
\]
We want to express it in the form \( A = (B)(C) \), where \( A \) is the original expression, \( B \) is the divisor, and \( C \) is the quotient.
None of the options you provided seem to match perfectly with a standard interpretation of the division as multiplication. However, the concept being illustrated is that you can express the division as a product where one factor is the reciprocal of the divisor.
So, if we consider:
\[
\frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right) (8x^2 - 4x + 12)
\]
This would correspond to a choice where:
\[
\frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right)(8x^2 - 4x + 12)
\]
Given your options, the one that correctly demonstrates this relationship is:
**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**
Which can be simplified further if needed to express how dividing by \( 4x \) can be thought of as multiplying by its reciprocal.