Factor over the Complex Numbers x^4-625

$x^{4} - 625$

Step 1

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

${(x^{2})}^{2} - 625$

Step 2

Rewrite $625$ as $25^{2}$ .

${(x^{2})}^{2} - 25^{2}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - i^{2} = (a + i) (a - i)$ where $a = x^{2}$ and $i = 25$ .

$(x^{2} + 25) (x^{2} - 25)$

Step 4

Simplify.

Rewrite $25$ as $5^{2}$ .

$(x^{2} + 25) (x^{2} - 5^{2})$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - i^{2} = (a + i) (a - i)$ where $a = x$ and $i = 5$ .

$(x^{2} + 25) ((x + 5) (x - 5))$

Remove unnecessary parentheses.

$(x^{2} + 25) (x + 5) (x - 5)$

Step 5

Multiply the constant in the polynomial $x^{2} + 25$ by $- i^{2}$ where $i^{2}$ is equal to $- 1$ .

$(x^{2} - 25 i^{2}) (x + 5) (x - 5)$

Step 6

Rewrite $25 i^{2}$ as ${(5 i)}^{2}$ .

$(x^{2} - {(5 i)}^{2}) (x + 5) (x - 5)$

Step 7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - i^{2} = (a + i) (a - i)$ where $a = x$ and $i = 5 i$ .

$(x + 5 i) (x - (5 i)) (x + 5) (x - 5)$

Step 8

Multiply $5$ by $- 1$ .

$(x + 5 i) (x - 5 i) (x + 5) (x - 5)$

Factor over the Complex Numbers x^4-625

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