To simplify the given quotient [tex]\(\frac{3x^2 - 27x}{2x^2 + 13x - 7} \div \frac{3x}{4x^2 - 1}\)[/tex], follow these steps:
1. Rewrite the Given Expression: Convert the division problem into a multiplication problem by multiplying by the reciprocal of the second fraction.
[tex]\[
\frac{3x^2 - 27x}{2x^2 + 13x - 7} \div \frac{3x}{4x^2 - 1} = \frac{3x^2 - 27x}{2x^2 + 13x - 7} \times \frac{4x^2 - 1}{3x}
\][/tex]
2. Factor the Expressions: Factorize the numerator and the denominator wherever possible.
- [tex]\(3x^2 - 27x = 3x(x - 9)\)[/tex]
- [tex]\(4x^2 - 1 = (2x - 1)(2x + 1)\)[/tex]
The middle expression [tex]\(2x^2 + 13x - 7\)[/tex] does not factor neatly, so keep it as is for now.
3. Write the Factored Form: Substitute the factors back into the expression.
[tex]\[
\frac{3x(x - 9)}{2x^2 + 13x - 7} \times \frac{(2x - 1)(2x + 1)}{3x}
\][/tex]
4. Cancel Common Terms: Identify and cancel out any common terms from the numerator and the denominator.
The factor [tex]\(3x\)[/tex] appears in both the numerator and the denominator and can be cancelled out.
[tex]\[
\frac{(x - 9)(2x - 1)(2x + 1)}{2x^2 + 13x - 7}
\][/tex]
Now the simplified form of the quotient is:
[tex]\[
\frac{(x - 9)(2x - 1)(2x + 1)}{2x^2 + 13x - 7}
\][/tex]
5. Determine when the Expression Does Not Exist: The expression does not exist where the denominator is zero. Therefore, solve:
[tex]\[
2x^2 + 13x - 7 = 0
\][/tex]
Using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex] where [tex]\(a = 2\)[/tex], [tex]\(b = 13\)[/tex], and [tex]\(c = -7\)[/tex]:
The roots are [tex]\(x = 0.5\)[/tex] and [tex]\(x = -7.0\)[/tex].
6. Inclusion of [tex]\(x = 0\)[/tex]: Since [tex]\(x = 0\)[/tex] also makes the original denominator [tex]\(3x\)[/tex] in the second fraction equal to zero, we must include this in the restrictions.
Hence, the simplest form of the numerator is [tex]\((x - 9)(2x - 1)(2x + 1)\)[/tex], and the expression does not exist when [tex]\(x = 0\)[/tex], [tex]\(x = 0.5\)[/tex], or [tex]\(x = -7.0\)[/tex].
So, select the correct answer from each drop-down menu:
- The simplest form of this quotient has a numerator of [tex]\((x - 9)(2x - 1)(2x + 1)\)[/tex]
- The expression does not exist when [tex]\(x = 0\)[/tex].
