Algebra Examples

$x^{4} + 41 x^{2} + 400$

Step 1

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

${(x^{2})}^{2} + 41 x^{2} + 400$

Step 2

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

$u^{2} + 41 u + 400$

Step 3

Factor $u^{2} + 41 u + 400$ 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 $400$ and whose sum is $41$ .

$16, 25$

Step 3.2

Write the factored form using these integers.

$(u + 16) (u + 25)$

Step 4

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

$(x^{2} + 16) (x^{2} + 25)$

Step 5

Factor over the complex numbers.

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Step 5.1

Multiply the constant in the polynomial $x^{2} + 16$ by $- i^{2}$ where $i^{2}$ is equal to $- 1$ .

$(x^{2} - 16 i^{2}) (x^{2} + 25)$

Step 5.2

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

$(x^{2} - 16 i^{2}) (x^{2} - 25 i^{2})$

Step 5.3

Rewrite $16 i^{2}$ as ${(4 i)}^{2}$ .

$(x^{2} - {(4 i)}^{2}) (x^{2} - 25 i^{2})$

Step 5.4

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 $i = 4 i$ .

$(x + 4 i) (x - (4 i)) (x^{2} - 25 i^{2})$

Step 5.5

Multiply $4$ by $- 1$ .

$(x + 4 i) (x - 4 i) (x^{2} - 25 i^{2})$

Step 5.6

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

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

Step 5.7

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 $i = 5 i$ .

$(x + 4 i) (x - 4 i) ((x + 5 i) (x - (5 i)))$

Step 5.8

Factor.

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Step 5.8.1

Multiply $5$ by $- 1$ .

$(x + 4 i) (x - 4 i) ((x + 5 i) (x - 5 i))$

Step 5.8.2

Remove unnecessary parentheses.

$(x + 4 i) (x - 4 i) (x + 5 i) (x - 5 i)$