Basic Math Examples

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

Step 1

Combine the numerators over the common denominator.

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

Step 2

Simplify the numerator.

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Step 2.1

Factor $3$ out of $3 z^{3} + 81$ .

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Step 2.1.1

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

$\frac{3 (z^{3}) + 81}{z^{2} - 3 z + 9}$

Step 2.1.2

Factor $3$ out of $81$ .

$\frac{3 z^{3} + 3 \cdot 27}{z^{2} - 3 z + 9}$

Step 2.1.3

Factor $3$ out of $3 z^{3} + 3 \cdot 27$ .

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

Step 2.2

Rewrite $27$ as $3^{3}$ .

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

Step 2.3

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

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

Step 2.4

Simplify.

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Step 2.4.1

Multiply $3$ by $- 1$ .

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

Step 2.4.2

Raise $3$ to the power of $2$ .

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

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $z^{2} - 3 z + 9$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Divide $3 (z + 3)$ by $1$ .

$3 (z + 3)$

Step 3.2

Apply the distributive property.

$3 z + 3 \cdot 3$

Step 3.3

Multiply $3$ by $3$ .

$3 z + 9$