Let's analyze the given quotient step-by-step.
Given:
[tex]\[ \frac{3 x^2 - 27 x}{2 x^2 + 13 x - 7} \div \frac{3 x}{4 x^2 - 1} \][/tex]
### Step 1: Simplify Each Fraction
First, simplify each fraction separately.
1. Simplify [tex]\(\frac{3 x^2 - 27 x}{2 x^2 + 13 x - 7}\)[/tex]:
- Numerator: [tex]\(3 x^2 - 27 x\)[/tex]
- Denominator: [tex]\(2 x^2 + 13 x - 7\)[/tex]
Factorize the numerator and denominator if possible:
- Numerator: [tex]\(3x(x - 9)\)[/tex]
- Denominator: This is a bit too complex for manual factorization here, so we will simplify it directly using algebraic techniques revealing simplified form stays as it is.
2. Simplify [tex]\(\frac{3 x}{4 x^2 - 1}\)[/tex]:
- Numerator: [tex]\(3 x\)[/tex]
- Denominator: [tex]\(4 x^2 - 1\)[/tex]
Factorize the denominator:
- Denominator: [tex]\((2x - 1)(2x + 1)\)[/tex]
### Step 2: Divide the Simplified Fractions
Dividing the two fractions is equivalent to multiplying the first fraction by the reciprocal of the second.
[tex]\[
\frac{3 x^2 - 27 x}{2 x^2 + 13 x - 7} \div \frac{3 x}{4 x^2 - 1} = \frac{3 x^2 - 27 x}{2 x^2 + 13 x - 7} \times \frac{4 x^2 - 1}{3 x}
\][/tex]
Combine into a single fraction:
[tex]\[
\frac{(3 x^2 - 27 x) \cdot (4 x^2 - 1)}{(2 x^2 + 13 x - 7) \cdot 3 x}
\][/tex]
### Step 3: Simplify the Resulting Expression
Simplify the fraction:
[tex]\[
\frac{(3 x^2 - 27 x) \cdot (4 x^2 - 1)}{(2 x^2 + 13 x - 7) \cdot 3 x}
\][/tex]
Factorize [tex]\(3 x^2 - 27 x\)[/tex] as [tex]\(3x(x - 9)\)[/tex].
Rewrite the entire fraction:
[tex]\[
\frac{3x(x - 9) \cdot (4 x^2 - 1)}{(2 x^2 + 13 x - 7) \cdot 3 x}
\][/tex]
Cancel out terms common to numerator and denominator:
[tex]\[
\frac{(x - 9) \cdot (4 x^2 - 1)}{2 x^2 + 13 x - 7}
\][/tex]
Notice the denominator [tex]\(4 x^2 - 1\)[/tex] can be written as [tex]\((2x - 1)(2x + 1)\)[/tex]:
Simplify further:
[tex]\[
= \frac{(x - 9) \cdot (2x - 1)(2x + 1)}{2 x^2 + 13 x - 7}
\][/tex]
### Step 4: Simplify to the Most Reduced Form
Combine and further reduce any additional common factors which simplification gives:
### Result
As we solved the simplification, we found the final form of numerator and denominator are :
[tex]\(\boxed{2x^2 - 17x - 9}\)[/tex]
and
[tex]\(\boxed{x + 7}\)[/tex]
### Step 5: Identify Non-existent Points
Finally, determine when the expression does not exist, which occurs when the denominator is zero.
[tex]\[
x + 7 = 0 \implies x = -7
\][/tex]
Thus, the expression does not exist for [tex]\(x = \boxed{-7}\)[/tex].
So, the simplest form of the quotient is:
- Numerator: [tex]\(2x^2 - 17x - 9\)[/tex]
- Denominator: [tex]\(x + 7\)[/tex]
- The expression does not exist when [tex]\(x = -7\)[/tex].
