Finite Math Examples

$\frac{4 x^{2} - 4 x - 3}{8 x^{3} + 1}$

Step 1

Factor by grouping.

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Step 1.1

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 $a \cdot c = 4 \cdot - 3 = - 12$ and whose sum is $b = - 4$ .

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Step 1.1.1

Factor $- 4$ out of $- 4 x$ .

$\frac{4 x^{2} - 4 (x) - 3}{8 x^{3} + 1}$

Step 1.1.2

Rewrite $- 4$ as $2$ plus $- 6$

$\frac{4 x^{2} + (2 - 6) x - 3}{8 x^{3} + 1}$

Step 1.1.3

Apply the distributive property.

$\frac{4 x^{2} + 2 x - 6 x - 3}{8 x^{3} + 1}$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$\frac{(4 x^{2} + 2 x) - 6 x - 3}{8 x^{3} + 1}$

Step 1.2.2

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

$\frac{2 x (2 x + 1) - 3 (2 x + 1)}{8 x^{3} + 1}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$\frac{(2 x + 1) (2 x - 3)}{8 x^{3} + 1}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $8 x^{3}$ as ${(2 x)}^{3}$ .

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

Step 2.2

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

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

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 $a = 2 x$ and $b = 1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the product rule to $2 x$ .

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

Step 2.4.2

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

$\frac{(2 x + 1) (2 x - 3)}{(2 x + 1) (4 x^{2} - (2 x) \cdot 1 + 1^{2})}$

Step 2.4.3

Multiply $2$ by $- 1$ .

$\frac{(2 x + 1) (2 x - 3)}{(2 x + 1) (4 x^{2} - 2 x \cdot 1 + 1^{2})}$

Step 2.4.4

Multiply $- 2$ by $1$ .

$\frac{(2 x + 1) (2 x - 3)}{(2 x + 1) (4 x^{2} - 2 x + 1^{2})}$

Step 2.4.5

One to any power is one.

$\frac{(2 x + 1) (2 x - 3)}{(2 x + 1) (4 x^{2} - 2 x + 1)}$

Step 3

Cancel the common factor of $2 x + 1$ .

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Step 3.1

Cancel the common factor.

$\frac{(2 x + 1) (2 x - 3)}{(2 x + 1) (4 x^{2} - 2 x + 1)}$

Step 3.2

Rewrite the expression.

$\frac{2 x - 3}{4 x^{2} - 2 x + 1}$