Precalculus Examples

$\frac{z^{3} \cdot 2 z^{2}}{z^{2} + 7 z - 18} = \frac{- 12 z - 27}{z + 9}$

Step 1

Simplify $\frac{z^{3} \cdot (2 z^{2})}{z^{2} + 7 z - 18}$ .

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Step 1.1

Multiply $z^{3}$ by $z^{2}$ by adding the exponents.

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Step 1.1.1

Move $z^{2}$ .

$\frac{z^{2} z^{3} \cdot 2}{z^{2} + 7 z - 18} = \frac{- 12 z - 27}{z + 9}$

Step 1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{z^{2 + 3} \cdot 2}{z^{2} + 7 z - 18} = \frac{- 12 z - 27}{z + 9}$

Step 1.1.3

Add $2$ and $3$ .

$\frac{z^{5} \cdot 2}{z^{2} + 7 z - 18} = \frac{- 12 z - 27}{z + 9}$

Step 1.2

Factor $z^{2} + 7 z - 18$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 18$ and whose sum is $7$ .

$- 2, 9$

Step 1.2.2

Write the factored form using these integers.

$\frac{z^{5} \cdot 2}{(z - 2) (z + 9)} = \frac{- 12 z - 27}{z + 9}$

Step 1.3

Move $2$ to the left of $z^{5}$ .

$\frac{2 z^{5}}{(z - 2) (z + 9)} = \frac{- 12 z - 27}{z + 9}$

Step 2

Simplify $\frac{- 12 z - 27}{z + 9}$ .

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Step 2.1

Factor $3$ out of $- 12 z - 27$ .

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Step 2.1.1

Factor $3$ out of $- 12 z$ .

$\frac{2 z^{5}}{(z - 2) (z + 9)} = \frac{3 (- 4 z) - 27}{z + 9}$

Step 2.1.2

Factor $3$ out of $- 27$ .

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

Step 2.1.3

Factor $3$ out of $3 (- 4 z) + 3 (- 9)$ .

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

Step 2.2

Factor $- 1$ out of $- 4 z$ .

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

Step 2.3

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{2 z^{5}}{(z - 2) (z + 9)} = \frac{3 (- (4 z) - 1 (9))}{z + 9}$

Step 2.4

Factor $- 1$ out of $- (4 z) - 1 (9)$ .

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

Step 2.5

Simplify the expression.

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Step 2.5.1

Rewrite $- (4 z + 9)$ as $- 1 (4 z + 9)$ .

$\frac{2 z^{5}}{(z - 2) (z + 9)} = \frac{3 (- 1 (4 z + 9))}{z + 9}$

Step 2.5.2

Move the negative in front of the fraction.

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

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$z \approx 1.57621015$

Step 4