Algebra Examples

$\frac{5 x^{8} - 20 x^{4}}{5 x^{8} + 20 x^{6} + 20 x^{4}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $5 x^{4}$ out of $5 x^{8} - 20 x^{4}$ .

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Step 1.1.1

Factor $5 x^{4}$ out of $5 x^{8}$ .

$\frac{5 x^{4} (x^{4}) - 20 x^{4}}{5 x^{8} + 20 x^{6} + 20 x^{4}}$

Step 1.1.2

Factor $5 x^{4}$ out of $- 20 x^{4}$ .

$\frac{5 x^{4} (x^{4}) + 5 x^{4} (- 4)}{5 x^{8} + 20 x^{6} + 20 x^{4}}$

Step 1.1.3

Factor $5 x^{4}$ out of $5 x^{4} (x^{4}) + 5 x^{4} (- 4)$ .

$\frac{5 x^{4} (x^{4} - 4)}{5 x^{8} + 20 x^{6} + 20 x^{4}}$

Step 1.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{5 x^{4} ({(x^{2})}^{2} - 4)}{5 x^{8} + 20 x^{6} + 20 x^{4}}$

Step 1.3

Rewrite $4$ as $2^{2}$ .

$\frac{5 x^{4} ({(x^{2})}^{2} - 2^{2})}{5 x^{8} + 20 x^{6} + 20 x^{4}}$

Step 1.4

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

$\frac{5 x^{4} (x^{2} + 2) (x^{2} - 2)}{5 x^{8} + 20 x^{6} + 20 x^{4}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $5 x^{4}$ out of $5 x^{8} + 20 x^{6} + 20 x^{4}$ .

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Step 2.1.1

Factor $5 x^{4}$ out of $5 x^{8}$ .

$\frac{5 x^{4} (x^{2} + 2) (x^{2} - 2)}{5 x^{4} (x^{4}) + 20 x^{6} + 20 x^{4}}$

Step 2.1.2

Factor $5 x^{4}$ out of $20 x^{6}$ .

$\frac{5 x^{4} (x^{2} + 2) (x^{2} - 2)}{5 x^{4} (x^{4}) + 5 x^{4} (4 x^{2}) + 20 x^{4}}$

Step 2.1.3

Factor $5 x^{4}$ out of $20 x^{4}$ .

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

Step 2.1.4

Factor $5 x^{4}$ out of $5 x^{4} (x^{4}) + 5 x^{4} (4 x^{2})$ .

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

Step 2.1.5

Factor $5 x^{4}$ out of $5 x^{4} (x^{4} + 4 x^{2}) + 5 x^{4} (4)$ .

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

Step 2.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 2.3

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$\frac{5 x^{4} (x^{2} + 2) (x^{2} - 2)}{5 x^{4} (u^{2} + 4 u + 4)}$

Step 2.4

Factor using the perfect square rule.

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Step 2.4.1

Rewrite $4$ as $2^{2}$ .

$\frac{5 x^{4} (x^{2} + 2) (x^{2} - 2)}{5 x^{4} (u^{2} + 4 u + 2^{2})}$

Step 2.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 u = 2 \cdot u \cdot 2$

Step 2.4.3

Rewrite the polynomial.

$\frac{5 x^{4} (x^{2} + 2) (x^{2} - 2)}{5 x^{4} (u^{2} + 2 \cdot u \cdot 2 + 2^{2})}$

Step 2.4.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 2$ .

$\frac{5 x^{4} (x^{2} + 2) (x^{2} - 2)}{5 x^{4} {(u + 2)}^{2}}$

Step 2.5

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $5$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x^{4}$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Cancel the common factors.

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Step 4.1

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

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

$\frac{x^{2} - 2}{x^{2} + 2}$