To solve the given problem, [tex]\(\frac{3z^2 - 2z - 8}{z - 6} \div \frac{z - 2}{z^2 - 9z + 18}\)[/tex], we need to follow the steps for dividing rational expressions.
### Step-by-Step Solution:
1. Rewrite the division as multiplication by the reciprocal:
When dividing by a fraction, it is equivalent to multiplying by the reciprocal of that fraction. So, we can rewrite our problem as follows:
[tex]\[
\frac{3z^2 - 2z - 8}{z - 6} \div \frac{z - 2}{z^2 - 9z + 18} = \frac{3z^2 - 2z - 8}{z - 6} \times \frac{z^2 - 9z + 18}{z - 2}
\][/tex]
2. Factor the polynomials if possible:
Let’s factorize the polynomials where possible.
- The numerator [tex]\(3z^2 - 2z - 8\)[/tex] is already in its simplest form (it doesn't factor neatly).
- The denominator [tex]\(z - 6\)[/tex] is also in its simplest form.
- The numerator [tex]\(z^2 - 9z + 18\)[/tex] can be factored as:
[tex]\[
z^2 - 9z + 18 = (z - 3)(z - 6)
\][/tex]
- The denominator [tex]\(z - 2\)[/tex] is in its simplest form.
3. Substitute the factored forms:
Now we rewrite the expression using the factored forms:
[tex]\[
\frac{3z^2 - 2z - 8}{z - 6} \times \frac{(z - 3)(z - 6)}{z - 2}
\][/tex]
4. Simplify the expression:
Cancel any common factors between the numerator and the denominator:
Notice that [tex]\((z - 6)\)[/tex] appears in both the numerator and the denominator, so it cancels out:
[tex]\[
\frac{3z^2 - 2z - 8}{1} \times \frac{(z - 3)}{z - 2} = (3z^2 - 2z - 8) \times \frac{z - 3}{z - 2}
\][/tex]
5. Perform the multiplication:
Multiply the remaining expressions:
[tex]\[
\frac{(3z^2 - 2z - 8)(z - 3)}{z - 2}
\][/tex]
Since there are no further common factors to cancel out in this particular problem, the simplified form of the given rational expressions after performing the multiplication is:
[tex]\[
3z^2 - 5z - 12
\][/tex]
### Final Answer:
[tex]\[
\boxed{3z^2 - 5z - 12}
\][/tex]
So, [tex]\(\frac{3z^2 - 2z - 8}{z - 6} \div \frac{z - 2}{z^2 - 9z + 18} = \frac{3z^2 - 2z - 8}{z - 6} \times \frac{z^2 - 9z + 18}{z - 2} = 3z^2 - 5z - 12\)[/tex].
