Certainly! Let's break down the given problem step by step:
1. Given Expression:
[tex]\[
\frac{3x^2 - 27x}{2x^2 + 13x - 7} \div \frac{3x}{4x^2 - 1}
\][/tex]
2. Simplification and Division:
To divide by a fraction, we multiply by its reciprocal. Therefore, we can rewrite the expression as:
[tex]\[
\frac{3x^2 - 27x}{2x^2 + 13x - 7} \times \frac{4x^2 - 1}{3x}
\][/tex]
3. Combine the Fractions:
We combine the fractions into a single fraction:
[tex]\[
\frac{(3x^2 - 27x)(4x^2 - 1)}{(2x^2 + 13x - 7)(3x)}
\][/tex]
4. Simplify the Expression:
Simplifying such algebraic expressions involves factorization and cancellation of common terms.
The simplest form of the given expression is found to have:
- Numerator: [tex]\( 2x^2 - 17x - 9 \)[/tex]
- Denominator: [tex]\( x + 7 \)[/tex]
5. Finding Points Where the Expression Does Not Exist:
The simplified expression does not exist where the denominator of the original or simplified expression is zero.
For the original denominator:
[tex]\[
(2x^2 + 13x - 7)(3x) = 0 \quad \text{(setting each part equal to zero)} \\
2x^2 + 13x - 7 = 0 \\
3x = 0
\][/tex]
For the factors in the expression, the critical points are:
[tex]\[
x = 0 \quad (\text{from } 3x = 0)
\][/tex]
Solving [tex]\( 2x^2 + 13x - 7 = 0 \)[/tex] would yield roots (we are given):
[tex]\[
x = -7 \quad \text{and} \quad x = \frac{1}{2}
\][/tex]
Therefore, we have:
- Simplified Numerator: [tex]\( 2x^2 - 17x - 9 \)[/tex]
- Points where the expression does not exist: [tex]\( x = -7, 0, \frac{1}{2} \)[/tex]
- Simplified Denominator: [tex]\( x + 7 \)[/tex]
Answer:
The simplest form of this quotient has a numerator of [tex]\(2x^2 - 17x - 9\)[/tex], the expression does not exist when [tex]\(x = -7, 0, \frac{1}{2}\)[/tex], and the denominator of [tex]\(x + 7\)[/tex].
