Finite Math Examples

$\frac{27 x^{3} - 1}{3 x - 1}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $27 x^{3}$ as ${(3 x)}^{3}$ .

$\frac{{(3 x)}^{3} - 1}{3 x - 1}$

Step 1.2

Rewrite $1$ as $1^{3}$ .

$\frac{{(3 x)}^{3} - 1^{3}}{3 x - 1}$

Step 1.3

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 $a = 3 x$ and $b = 1$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Apply the product rule to $3 x$ .

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

Step 1.4.2

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

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

Step 1.4.3

Multiply $3$ by $1$ .

$\frac{(3 x - 1) (9 x^{2} + 3 x + 1^{2})}{3 x - 1}$

Step 1.4.4

One to any power is one.

$\frac{(3 x - 1) (9 x^{2} + 3 x + 1)}{3 x - 1}$

Step 2

Cancel the common factor of $3 x - 1$ .

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Step 2.1

Cancel the common factor.

$\frac{(3 x - 1) (9 x^{2} + 3 x + 1)}{3 x - 1}$

Step 2.2

Divide $9 x^{2} + 3 x + 1$ by $1$ .

$9 x^{2} + 3 x + 1$