Algebra Examples

$x^{4} + 170 x^{2} + 169$

Step 1

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

${(x^{2})}^{2} + 170 x^{2} + 169$

Step 2

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

$u^{2} + 170 u + 169$

Step 3

Factor $u^{2} + 170 u + 169$ using the AC method.

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Step 3.1

Consider the form $x^{2} + i x + c$ . Find a pair of integers whose product is $c$ and whose sum is $i$ . In this case, whose product is $169$ and whose sum is $170$ .

$1, 169$

Step 3.2

Write the factored form using these integers.

$(u + 1) (u + 169)$

Step 4

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

$(x^{2} + 1) (x^{2} + 169)$

Step 5

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

$(x^{2} - 1 i^{2}) (x^{2} + 169)$

Step 6

Multiply the constant in the polynomial $x^{2} + 169$ by $- i^{2}$ where $i^{2}$ is equal to $- 1$ .

$(x^{2} - 1 i^{2}) (x^{2} - 169 i^{2})$

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 $i = i$ .

$(x + i) (x - i) (x^{2} - 169 i^{2})$

Step 8

Rewrite $169 i^{2}$ as ${(13 i)}^{2}$ .

$(x + i) (x - i) (x^{2} - {(13 i)}^{2})$

Step 9

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 $i = 13 i$ .

$(x + i) (x - i) ((x + 13 i) (x - (13 i)))$

Step 10

Factor.

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Step 10.1

Multiply $13$ by $- 1$ .

$(x + i) (x - i) ((x + 13 i) (x - 13 i))$

Step 10.2

Remove unnecessary parentheses.

$(x + i) (x - i) (x + 13 i) (x - 13 i)$