Finite Math Examples

$\frac{12 x^{2} + 2 x - 2}{27 x^{3} - 1}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $2$ out of $12 x^{2} + 2 x - 2$ .

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Step 1.1.1

Factor $2$ out of $12 x^{2}$ .

$\frac{2 (6 x^{2}) + 2 x - 2}{27 x^{3} - 1}$

Step 1.1.2

Factor $2$ out of $2 x$ .

$\frac{2 (6 x^{2}) + 2 (x) - 2}{27 x^{3} - 1}$

Step 1.1.3

Factor $2$ out of $- 2$ .

$\frac{2 (6 x^{2}) + 2 (x) + 2 (- 1)}{27 x^{3} - 1}$

Step 1.1.4

Factor $2$ out of $2 (6 x^{2}) + 2 (x)$ .

$\frac{2 (6 x^{2} + x) + 2 (- 1)}{27 x^{3} - 1}$

Step 1.1.5

Factor $2$ out of $2 (6 x^{2} + x) + 2 (- 1)$ .

$\frac{2 (6 x^{2} + x - 1)}{27 x^{3} - 1}$

Step 1.2

Factor by grouping.

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Step 1.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 6 \cdot - 1 = - 6$ and whose sum is $b = 1$ .

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Step 1.2.1.1

Multiply by $1$ .

$\frac{2 (6 x^{2} + 1 x - 1)}{27 x^{3} - 1}$

Step 1.2.1.2

Rewrite $1$ as $- 2$ plus $3$

$\frac{2 (6 x^{2} + (- 2 + 3) x - 1)}{27 x^{3} - 1}$

Step 1.2.1.3

Apply the distributive property.

$\frac{2 (6 x^{2} - 2 x + 3 x - 1)}{27 x^{3} - 1}$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{2 ((6 x^{2} - 2 x) + 3 x - 1)}{27 x^{3} - 1}$

Step 1.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{2 (2 x (3 x - 1) + 1 (3 x - 1))}{27 x^{3} - 1}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$\frac{2 ((3 x - 1) (2 x + 1))}{27 x^{3} - 1}$

$\frac{2 (3 x - 1) (2 x + 1)}{27 x^{3} - 1}$

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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 $b = 1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the product rule to $3 x$ .

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

Step 2.4.2

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

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

Step 2.4.3

Multiply $3$ by $1$ .

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

Step 2.4.4

One to any power is one.

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

Step 3

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

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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