To solve the given problem:
[tex]\[
\frac{3 z^2}{z^2-6 z} \div \frac{6 z^2-42 z}{z^2-10 z+21},
\][/tex]
we will follow these steps:
1. Rewrite the division as a multiplication by the reciprocal:
[tex]\[
\frac{3 z^2}{z^2-6 z} \cdot \frac{z^2-10 z+21}{6 z^2-42 z}.
\][/tex]
2. Factor the expressions wherever possible:
For the first denominator [tex]\(z^2 - 6z\)[/tex], factor out a [tex]\(z\)[/tex]:
[tex]\[
z^2 - 6z = z(z-6).
\][/tex]
For the second numerator [tex]\(z^2 - 10z + 21\)[/tex], factor it into two binomials:
[tex]\[
z^2 - 10z + 21 = (z-3)(z-7).
\][/tex]
For the second denominator [tex]\(6z^2 - 42z\)[/tex], factor out the greatest common factor (GCF), which is [tex]\(6z\)[/tex]:
[tex]\[
6z^2 - 42z = 6z(z-7).
\][/tex]
Now, substitute these factored forms back into the expression:
[tex]\[
\frac{3z^2}{z(z-6)} \cdot \frac{(z-3)(z-7)}{6z(z-7)}.
\][/tex]
3. Simplify the expression:
Cancel out common factors in the numerator and the denominator. Specifically, [tex]\(z\)[/tex] and [tex]\(z-7\)[/tex] appear in both the numerators and denominators, so they can be canceled:
[tex]\[
\frac{3z^2}{z(z-6)} \cdot \frac{(z-3)(z-7)}{6z(z-7)} = \frac{3z \cdot z}{z \cdot (z-6)} \cdot \frac{(z-3)}{6z}.
\][/tex]
[tex]\[
= \frac{3 \cdot z \cdot (z-3)}{6z(z-6)}.
\][/tex]
Next, cancel [tex]\(3z\)[/tex] from the numerator and the denominator:
[tex]\[
= \frac{(z-3)}{2(z-6)}.
\][/tex]
4. Result:
The final simplified expression is:
[tex]\[
\frac{(z-3)}{2(z-6)}.
\][/tex]
This is the completely simplified rational expression for the given problem.
