Simplify (5x^8-20x^4)/(5x^8+20x^6+20x^4)

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

Simplify the numerator.

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

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

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

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

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

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

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

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

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

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

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

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 $b = 2$ .

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

Simplify the denominator.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Factor using the perfect square rule.

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

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

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

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

Rewrite the polynomial.

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

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

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

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

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

Reduce the expression by cancelling the common factors.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (5x^8-20x^4)/(5x^8+20x^6+20x^4)

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