Sure, let's break this problem down step by step.
Given the quotient:
[tex]\[
\frac{3 x^2 - 27 x}{2 x^3 + 13 x - 7} \div \frac{3 x}{4 x^3 - 1}
\][/tex]
First, rewrite the division as multiplication by the reciprocal:
[tex]\[
\frac{3 x^2 - 27 x}{2 x^3 + 13 x - 7} \times \frac{4 x^3 - 1}{3 x}
\][/tex]
Next, multiply the numerators and the denominators:
[tex]\[
\frac{(3 x^2 - 27 x) \times (4 x^3 - 1)}{(2 x^3 + 13 x - 7) \times (3 x)}
\][/tex]
To simplify the quotient, we factor the expressions and find common factors.
The numerator and the denominator of the simplified expression will be:
Numerator: [tex]\((x - 9) \times (4 x^3 - 1)\)[/tex]
Denominator: [tex]\(2 x^3 + 13 x - 7\)[/tex]
In the simplified form, the quotient does not exist when the denominator is zero.
To determine the values of [tex]\(x\)[/tex] that make the denominator zero, we solve:
[tex]\[2 x^3 + 13 x - 7 = 0\][/tex]
The solutions to this are:
[tex]\[
\left\{-\frac{13}{6\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}} + \left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}, -\frac{\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}}{2} + \frac{13}{12\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}} + I\left(\frac{13\sqrt{3}}{12\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}} + \frac{\sqrt{3}\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}}{2}\right), -\frac{\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}}{2} + \frac{13}{12\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}} + I\left(-\frac{\sqrt{3}\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}}{2} - \frac{13\sqrt{3}}{12\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}}\right)\right\}
\][/tex]
These values of [tex]\(x\)[/tex] will make the whole expression undefined.
Therefore, the answer to the question is:
The simplest form of the quotient has a numerator of:
[tex]\[
(x - 9) \times (4 x^3 - 1)
\][/tex]
The expression does not exist when [tex]\(x =\)[/tex]:
[tex]\[
\left\{-\frac{13}{6\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}} + \left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}, -\frac{\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}}{2} + \frac{13}{12\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}} + I\left(\frac{13\sqrt{3}}{12\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}} + \frac{\sqrt{3}\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}}{2}\right), -\frac{\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}}{2} + \frac{13}{12\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}} + I\left(-\frac{\sqrt{3}\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}}{2} - \frac{13\sqrt{3}}{12\left(\frac{7}{4} + \frac{\sqrt{17151}}{36}\right)^{\frac{1}{3}}}\right)\right\}
\][/tex]
And a denominator of:
[tex]\[
2 x^3 + 13 x - 7
\][/tex]
By matching the correct terms in the given dropdown sequence, you can accurately fill out the required answer.
