Finite Math Examples

$f (x) = \frac{x^{3} - 729}{x^{2} - 81}$

Step 1

Rewrite the function as an equation.

$y = \frac{x^{3} - 729}{x^{2} - 81}$

Step 2

Simplify $\frac{x^{3} - 729}{x^{2} - 81}$ .

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Rewrite $729$ as $9^{3}$ .

$y = \frac{x^{3} - 9^{3}}{x^{2} - 81}$

Step 2.1.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 9$ .

$y = \frac{(x - 9) (x^{2} + x \cdot 9 + 9^{2})}{x^{2} - 81}$

Step 2.1.3

Simplify.

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Step 2.1.3.1

Move $9$ to the left of $x$ .

$y = \frac{(x - 9) (x^{2} + 9 \cdot x + 9^{2})}{x^{2} - 81}$

Step 2.1.3.2

Raise $9$ to the power of $2$ .

$y = \frac{(x - 9) (x^{2} + 9 x + 81)}{x^{2} - 81}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Rewrite $81$ as $9^{2}$ .

$y = \frac{(x - 9) (x^{2} + 9 x + 81)}{x^{2} - 9^{2}}$

Step 2.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 9$ .

$y = \frac{(x - 9) (x^{2} + 9 x + 81)}{(x + 9) (x - 9)}$

Step 2.3

Cancel the common factor of $x - 9$ .

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Step 2.3.1

Cancel the common factor.

$y = \frac{(x - 9) (x^{2} + 9 x + 81)}{(x + 9) (x - 9)}$

Step 2.3.2

Rewrite the expression.

$y = \frac{x^{2} + 9 x + 81}{x + 9}$