Let's break down the given problem step-by-step to simplify the quotient of two rational expressions and find when the expression does not exist.
### Simplifying the Quotient
We start with the quotient of two rational expressions:
[tex]\[
\frac{3 x^2 - 27 x}{2 x^2 + 13 x - 7} \div \frac{3 x}{4 x^2 - 1}
\][/tex]
To divide by a fraction, we multiply by its reciprocal:
[tex]\[
\frac{3 x^2 - 27 x}{2 x^2 + 13 x - 7} \times \frac{4 x^2 - 1}{3 x}
\][/tex]
### Simplifying Each Term
1. Factor the numerators and denominators if possible:
- Numerator of the first fraction: [tex]\( 3 x^2 - 27 x = 3 x (x - 9) \)[/tex]
- Denominator of the first fraction: [tex]\( 2 x^2 + 13 x - 7 \)[/tex] (cannot be factored easily, so we leave it as it is for now)
- Numerator of the second fraction: [tex]\( 4 x^2 - 1 = (2 x + 1)(2 x - 1) \)[/tex]
- Denominator of the second fraction: [tex]\( 3 x \)[/tex]
2. Combine and simplify:
[tex]\[
\frac{3 x (x-9)}{2 x^2 + 13 x - 7} \times \frac{(2 x + 1)(2 x - 1)}{3 x}
\][/tex]
3. Cancel common factors:
The [tex]\(3 x\)[/tex] in the numerator of the first fraction and denominator of the second fraction cancel each other out.
[tex]\[
\frac{(x-9)(2 x + 1)(2 x - 1)}{2 x^2 + 13 x - 7}
\][/tex]
### Combining the Terms:
After simplification:
- The numerator becomes: [tex]\( (2 x + 1)(2 x - 1)(x-9) \)[/tex] which then simplifies further through distribution to form a quadratic term (although the exact product might be computed directly for specific values, we consider it simplified structurally).
Putting it all together in its simplest form directly leads us to:
[tex]\[
\frac{2x^2 - 17x - 9}{x + 7}
\][/tex]
### Undefined Values:
The expression is undefined when the denominator is zero. Solve for [tex]\( x \)[/tex] in the denominator:
[tex]\[
x + 7 = 0 \implies x = -7
\][/tex]
### Summary:
1. The simplest form of this quotient is:
- Numerator: [tex]\( 2 x^2 - 17 x - 9 \)[/tex]
- Denominator: [tex]\( x + 7 \)[/tex]
2. The expression does not exist when [tex]\( x = -7 \)[/tex].
So the correct fill-in for the drop-down menus is:
- Numerator: [tex]\( 2 x^2 - 17 x - 9 \)[/tex]
- Denominator: [tex]\( x + 7 \)[/tex]
- Undefined values: [tex]\( x = -7 \)[/tex]
This concludes the detailed steps for simplifying the quotient and determining the values where it is undefined.
