Basic Math Examples

$\frac{3 z^{3} - 27 z}{3 - z}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $3 z$ out of $3 z^{3} - 27 z$ .

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Step 1.1.1

Factor $3 z$ out of $3 z^{3}$ .

$\frac{3 z (z^{2}) - 27 z}{3 - z}$

Step 1.1.2

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

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

Step 1.1.3

Factor $3 z$ out of $3 z (z^{2}) + 3 z (- 9)$ .

$\frac{3 z (z^{2} - 9)}{3 - z}$

Step 1.2

Rewrite $9$ as $3^{2}$ .

$\frac{3 z (z^{2} - 3^{2})}{3 - z}$

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = z$ and $b = 3$ .

$\frac{3 z (z + 3) (z - 3)}{3 - z}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $z - 3$ and $3 - z$ .

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Step 2.1.1

Factor $- 1$ out of $z$ .

$\frac{3 z (z + 3) (- 1 (- z) - 3)}{3 - z}$

Step 2.1.2

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{3 z (z + 3) (- 1 (- z) - 1 (3))}{3 - z}$

Step 2.1.3

Factor $- 1$ out of $- 1 (- z) - 1 (3)$ .

$\frac{3 z (z + 3) (- 1 (- z + 3))}{3 - z}$

Step 2.1.4

Reorder terms.

$\frac{3 z (z + 3) (- 1 (- z + 3))}{- z + 3}$

Step 2.1.5

Cancel the common factor.

$\frac{3 z (z + 3) (- 1 (- z + 3))}{- z + 3}$

Step 2.1.6

Divide $3 z (z + 3) \cdot (- 1)$ by $1$ .

$3 z (z + 3) \cdot (- 1)$

Step 2.2

Apply the distributive property.

$(3 z \cdot z + 3 z \cdot 3) \cdot - 1$

Step 3

Multiply $z$ by $z$ by adding the exponents.

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Step 3.1

Move $z$ .

$(3 (z \cdot z) + 3 z \cdot 3) \cdot - 1$

Step 3.2

Multiply $z$ by $z$ .

$(3 z^{2} + 3 z \cdot 3) \cdot - 1$

Step 4

Multiply $3$ by $3$ .

$(3 z^{2} + 9 z) \cdot - 1$

Step 5

Apply the distributive property.

$3 z^{2} \cdot - 1 + 9 z \cdot - 1$

Step 6

Multiply.

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Step 6.1

Multiply $- 1$ by $3$ .

$- 3 z^{2} + 9 z \cdot - 1$

Step 6.2

Multiply $- 1$ by $9$ .

$- 3 z^{2} - 9 z$