Write as a Set of Linear Factors f(y)=y^4-625

$f (y) = y^{4} - 625$

Step 1

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

$f (y) = {(y^{2})}^{2} - 625$

Step 2

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

$f (y) = {(y^{2})}^{2} - 25^{2}$

Step 3

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

$f (y) = (y^{2} + 25) (y^{2} - 25)$

Step 4

Simplify.

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

$f (y) = (y^{2} + 25) (y^{2} - 5^{2})$

Factor.

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

$f (y) = (y^{2} + 25) ((y + 5) (y - 5))$

Remove unnecessary parentheses.

$f (y) = (y^{2} + 25) (y + 5) (y - 5)$

Step 5

Factor over the complex numbers.

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

$f (y) = (y^{2} - 25 i^{2}) (y + 5) (y - 5)$

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

$f (y) = (y^{2} - {(5 i)}^{2}) (y + 5) (y - 5)$

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

$f (y) = (y + 5 i) (y - (5 i)) (y + 5) (y - 5)$

Multiply $5$ by $- 1$ .

$f (y) = (y + 5 i) (y - 5 i) (y + 5) (y - 5)$

Write as a Set of Linear Factors f(y)=y^4-625

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