$$ \begin{array}{l} ext { Andrea }\ \checkmark 22\ 23\ 24\ 25\ 26\ imes 27\ 28\ 29\ 30\ ext { Espanol }\ 33\ ext { Divide the rational expressions. }\ \frac{z}{z-9} \div \frac{z^{3}+5 z^{2}-36 z}{z^{2}-81}= \end{array} $$

Question

$$ \begin{array}{l} 	ext { Andrea }\ \checkmark 22\ 23\ 24\ 25\ 26\ 	imes 27\ 28\ 29\ 30\ 	ext { Espanol }\ 33\ 	ext { Divide the rational expressions. }\ \frac{z}{z-9} \div \frac{z^{3}+5 z^{2}-36 z}{z^{2}-81}= \end{array} $$

UpStudy AI · Accepted Answer

To divide the given rational expressions:

$$
\frac{z}{z - 9} \div \frac{z^{3} + 5z^{2} - 36z}{z^{2} - 81}
$$

**Follow these steps to simplify:**

### 1. Rewrite the Division as Multiplication by the Reciprocal

Dividing by a fraction is the same as multiplying by its reciprocal. So,

$$
\frac{z}{z - 9} \div \frac{z^{3} + 5z^{2} - 36z}{z^{2} - 81} = \frac{z}{z - 9} \times \frac{z^{2} - 81}{z^{3} + 5z^{2} - 36z}
$$

### 2. Factor All Possible Expressions

Let's factor the numerator and the denominator wherever possible.

- **Factor $$ z^{2} - 81 $$:**

This is a difference of squares.

$$
z^{2} - 81 = (z - 9)(z + 9)
$$

- **Factor $$ z^{3} + 5z^{2} - 36z $$:**

Factor out the greatest common factor (GCF), which is $$ z $$:

$$
z^{3} + 5z^{2} - 36z = z(z^{2} + 5z - 36)
$$

Now, factor the quadratic $$ z^{2} + 5z - 36 $$:

$$
z^{2} + 5z - 36 = (z + 9)(z - 4)
$$

So,

$$
z^{3} + 5z^{2} - 36z = z(z + 9)(z - 4)
$$

### 3. Substitute the Factored Forms Back into the Expression

$$
\frac{z}{z - 9} \times \frac{(z - 9)(z + 9)}{z(z + 9)(z - 4)}
$$

### 4. Cancel Common Factors

Identify and cancel out the common terms in the numerator and the denominator.

- **Cancel $$ z $$:**

$$
\frac{\cancel{z}}{z - 9} \times \frac{(z - 9)(z + 9)}{\cancel{z}(z + 9)(z - 4)} = \frac{(z - 9)(z + 9)}{(z - 9)(z + 9)(z - 4)}
$$

- **Cancel $$ z - 9 $$:**

$$
\frac{\cancel{(z - 9)}(z + 9)}{\cancel{(z - 9)}(z + 9)(z - 4)} = \frac{z + 9}{(z + 9)(z - 4)}
$$

- **Cancel $$ z + 9 $$:**

$$
\frac{\cancel{z + 9}}{\cancel{z + 9}(z - 4)} = \frac{1}{z - 4}
$$

### 5. Final Simplified Expression

$$
\frac{z}{z - 9} \div \frac{z^{3} + 5z^{2} - 36z}{z^{2} - 81} = \frac{1}{z - 4}
$$

**Answer:**

$$
\frac{1}{z - 4}
$$
The simplified expression is $$ \frac{1}{z - 4} $$.

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Algebra Questions