To simplify the given quotient and find where it does not exist, we need to carry out the division by multiplying by the reciprocal. Here are the steps:
1. Rewrite the division as a multiplication by the reciprocal:
[tex]\[
\frac{3 x^2 - 37 x}{2 x^2 + 18 x - 7} \div \frac{3 x}{4 x^2 - 1} = \frac{3 x^2 - 37 x}{2 x^2 + 18 x - 7} \times \frac{4 x^2 - 1}{3 x}
\][/tex]
2. Simplify the resulting expression:
[tex]\[
\frac{(3 x^2 - 37 x)(4 x^2 - 1)}{(2 x^2 + 18 x - 7)(3 x)}
\][/tex]
3. Factor the polynomials where possible:
- [tex]\(3 x^2 - 37 x\)[/tex] can be factored as [tex]\( x(3x - 37) \)[/tex]
- [tex]\(4 x^2 - 1\)[/tex] is a difference of squares and can be factored as [tex]\((2x + 1)(2x - 1)\)[/tex]
- [tex]\(2 x^2 + 18 x - 7\)[/tex] is a bit more complex to factor, but we'll look for factors [tex]\((2x + a)(x + b)\)[/tex] that satisfy the equation. But for the given scope, you might need additional factor techniques or let’s assume it's in the simplest form as is.
4. Substituting these factorizations back:
[tex]\[
\frac{x(3x - 37)(2x + 1)(2x - 1)}{(2 x^2 + 18 x - 7)(3 x)}
\][/tex]
5. Cancel any common factors in the numerator and the denominator:
- The [tex]\(x\)[/tex] in [tex]\(x(3x - 37)\)[/tex] cancels out with [tex]\(3 x\)[/tex].
- Other factors might need polynomial solutions approach but we could try simpler arithmetic based reduction.
So the expression becomes:
[tex]\[
\frac{(3x - 37)(2x + 1)(2x - 1)}{2 x^2 + 18 x - 7}
\][/tex]
6. Find the conditions where the expression does not exist:
- The expression does not exist where the denominator is zero.
[tex]\[
3x = 0 \implies x = 0
\][/tex]
Also, look for values where the denominator [tex]\(2 x^2 + 18 x - 7 = 0\)[/tex]
Solve the quadratic equation [tex]\( 2x^2 + 18x - 7 = 0\)[/tex] using quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \implies b=18, a=2, c=-7
\][/tex]
yielding:
[tex]\[
x = \frac{18 \pm \sqrt{324 + 56}}{4} \implies \frac{18 \pm \sqrt{380}}{4}
\][/tex]
finding roots appropriately [tex]\(\frac{9 \pm \sqrt{95}}{2}\)[/tex].
Therefore, the simplified form numerator, non-existing points and denominator can be determined as follows:
- The simplest numerator form is [tex]\( (3x - 37)(2x + 1)(2x - 1) \)[/tex]
- The denominator is [tex]\( 2 x^2 + 18 x - 7 \)[/tex]
- The expression does not exist when [tex]\( x = 0 \)[/tex] or [tex]\( x = \frac{9 \pm \sqrt{95}}{2} \)[/tex].
So choose the options from drop-down as per solution enhancing simplification.
