Finite Math Examples

$\frac{2 n^{8} + 4 n^{5} - 6 n^{2}}{- 2 n^{4}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $2 n^{2}$ out of $2 n^{8} + 4 n^{5} - 6 n^{2}$ .

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Step 1.1.1

Factor $2 n^{2}$ out of $2 n^{8}$ .

$\frac{2 n^{2} (n^{6}) + 4 n^{5} - 6 n^{2}}{- 2 n^{4}}$

Step 1.1.2

Factor $2 n^{2}$ out of $4 n^{5}$ .

$\frac{2 n^{2} (n^{6}) + 2 n^{2} (2 n^{3}) - 6 n^{2}}{- 2 n^{4}}$

Step 1.1.3

Factor $2 n^{2}$ out of $- 6 n^{2}$ .

$\frac{2 n^{2} (n^{6}) + 2 n^{2} (2 n^{3}) + 2 n^{2} (- 3)}{- 2 n^{4}}$

Step 1.1.4

Factor $2 n^{2}$ out of $2 n^{2} (n^{6}) + 2 n^{2} (2 n^{3})$ .

$\frac{2 n^{2} (n^{6} + 2 n^{3}) + 2 n^{2} (- 3)}{- 2 n^{4}}$

Step 1.1.5

Factor $2 n^{2}$ out of $2 n^{2} (n^{6} + 2 n^{3}) + 2 n^{2} (- 3)$ .

$\frac{2 n^{2} (n^{6} + 2 n^{3} - 3)}{- 2 n^{4}}$

Step 1.2

Rewrite $n^{6}$ as ${(n^{3})}^{2}$ .

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

Step 1.3

Let $u = n^{3}$ . Substitute $u$ for all occurrences of $n^{3}$ .

$\frac{2 n^{2} (u^{2} + 2 u - 3)}{- 2 n^{4}}$

Step 1.4

Factor $u^{2} + 2 u - 3$ using the AC method.

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Step 1.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 1.4.2

Write the factored form using these integers.

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

Step 1.5

Replace all occurrences of $u$ with $n^{3}$ .

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

Step 1.6

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

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

Step 1.7

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 $a = n$ and $b = 1$ .

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

Step 1.8

Simplify.

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Step 1.8.1

Multiply $n$ by $1$ .

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

Step 1.8.2

One to any power is one.

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

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

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $2$ and $- 2$ .

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Step 2.1.1

Factor $2$ out of $2 n^{2} (n - 1) (n^{2} + n + 1) (n^{3} + 3)$ .

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $2$ out of $- 2 n^{4}$ .

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

Step 2.1.2.2

Cancel the common factor.

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

Step 2.1.2.3

Rewrite the expression.

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

Step 2.2

Cancel the common factor of $n^{2}$ and $n^{4}$ .

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Step 2.2.1

Factor $n^{2}$ out of $((n^{2} (n - 1)) (n^{2} + n + 1)) (n^{3} + 3)$ .

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

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $n^{2}$ out of $- n^{4}$ .

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 2.3

Move the negative in front of the fraction.

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