To divide the given rational expressions, we follow a series of steps to simplify the expression completely. Let's rewrite the division as multiplication by the reciprocal and then simplify step-by-step.
Given:
[tex]\[ \frac{3z^2}{z^2 - 6z} \div \frac{6z^2 - 42z}{z^2 - 10z + 21} \][/tex]
This can be rewritten as:
[tex]\[ \frac{3z^2}{z^2 - 6z} \times \frac{z^2 - 10z + 21}{6z^2 - 42z} \][/tex]
Now we perform the multiplication:
[tex]\[ \frac{3z^2}{z^2 - 6z} \times \frac{z^2 - 10z + 21}{6z^2 - 42z} = \frac{3z^2 (z^2 - 10z + 21)}{(z^2 - 6z)(6z^2 - 42z)} \][/tex]
Next, we need to factor each polynomial in the expression:
1. Factor [tex]\( z^2 - 6z \)[/tex]:
[tex]\[ z^2 - 6z = z(z - 6) \][/tex]
2. Factor [tex]\( 6z^2 - 42z \)[/tex]:
[tex]\[ 6z^2 - 42z = 6z(z - 7) \][/tex]
3. Factor [tex]\( z^2 - 10z + 21 \)[/tex]:
[tex]\[ z^2 - 10z + 21 = (z - 3)(z - 7) \][/tex]
Using these factorizations, the expression now becomes:
[tex]\[ \frac{3z^2 \cdot (z - 3)(z - 7)}{z(z - 6) \cdot 6z(z - 7)} \][/tex]
Simplifying, we can cancel common factors in the numerator and the denominator:
[tex]\[ \frac{3z^2 \cdot (z - 3) \cdot (z - 7)}{6z^2 \cdot (z - 6) \cdot (z - 7)} \][/tex]
Cancel [tex]\( z^2 \)[/tex] and [tex]\( (z - 7) \)[/tex] from the numerator and the denominator:
[tex]\[ \frac{3 \cdot (z - 3)}{6 \cdot (z - 6)} \][/tex]
Now simplify the constants present:
[tex]\[ \frac{3 \cdot (z - 3)}{6 \cdot (z - 6)} = \frac{(z - 3)}{2 \cdot (z - 6)} \][/tex]
Hence, the completely simplified rational expression is:
[tex]\[ \frac{z - 3}{2(z - 6)} \][/tex]
This is the final simplified form of the expression given in the problem.
