Algebra Examples

$f (x) = x^{4} + 5 x^{2} + 4$

Step 1

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

$f (x) = {(x^{2})}^{2} + 5 x^{2} + 4$

Step 2

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

$f (x) = u^{2} + 5 u + 4$

Step 3

Factor $u^{2} + 5 u + 4$ using the AC method.

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Step 3.1

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 $4$ and whose sum is $5$ .

$1, 4$

Step 3.2

Write the factored form using these integers.

$f (x) = (u + 1) (u + 4)$

Step 4

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

$f (x) = (x^{2} + 1) (x^{2} + 4)$

Step 5

Factor over the complex numbers.

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Step 5.1

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

$f (x) = (x^{2} - 1 i^{2}) (x^{2} + 4)$

Step 5.2

Multiply the constant in the polynomial $x^{2} + 4$ by $- i^{2}$ where $i^{2}$ is equal to $- 1$ .

$f (x) = (x^{2} - 1 i^{2}) (x^{2} - 4 i^{2})$

Step 5.3

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

$f (x) = (x + i) (x - i) (x^{2} - 4 i^{2})$

Step 5.4

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

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

Step 5.5

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 $i = 2 i$ .

$f (x) = (x + i) (x - i) ((x + 2 i) (x - (2 i)))$

Step 5.6

Factor.

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Step 5.6.1

Multiply $2$ by $- 1$ .

$f (x) = (x + i) (x - i) ((x + 2 i) (x - 2 i))$

Step 5.6.2

Remove unnecessary parentheses.

$f (x) = (x + i) (x - i) (x + 2 i) (x - 2 i)$