Algebra Examples

$x^{8} - 5 x^{4} + 4 = 0$

Step 1

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

${(x^{4})}^{2} - 5 x^{4} + 4 = 0$

Step 2

Let $u = x^{4}$ . Substitute $u$ for all occurrences of $x^{4}$ .

$u^{2} - 5 u + 4 = 0$

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 $- 5$ .

$- 4, - 1$

Step 3.2

Write the factored form using these integers.

$(u - 4) (u - 1) = 0$

Step 4

Replace all occurrences of $u$ with $x^{4}$ .

$(x^{4} - 4) (x^{4} - 1) = 0$

Step 5

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

$({(x^{2})}^{2} - 4) (x^{4} - 1) = 0$

Step 6

Rewrite $4$ as $2^{2}$ .

$({(x^{2})}^{2} - 2^{2}) (x^{4} - 1) = 0$

Step 7

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

$(x^{2} + 2) (x^{2} - 2) (x^{4} - 1) = 0$

Step 8

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

$(x^{2} + 2) (x^{2} - 2) ({(x^{2})}^{2} - 1) = 0$

Step 9

Rewrite $1$ as $1^{2}$ .

$(x^{2} + 2) (x^{2} - 2) ({(x^{2})}^{2} - 1^{2}) = 0$

Step 10

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 1$ .

$(x^{2} + 2) (x^{2} - 2) ((x^{2} + 1) (x^{2} - 1)) = 0$

Step 11

Factor.

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Step 11.1

Simplify.

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Step 11.1.1

Rewrite $1$ as $1^{2}$ .

$(x^{2} + 2) (x^{2} - 2) ((x^{2} + 1) (x^{2} - 1^{2})) = 0$

Step 11.1.2

Factor.

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Step 11.1.2.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$(x^{2} + 2) (x^{2} - 2) ((x^{2} + 1) ((x + 1) (x - 1))) = 0$

Step 11.1.2.2

Remove unnecessary parentheses.

$(x^{2} + 2) (x^{2} - 2) ((x^{2} + 1) (x + 1) (x - 1)) = 0$

Step 11.2

Remove unnecessary parentheses.

$(x^{2} + 2) (x^{2} - 2) (x^{2} + 1) (x + 1) (x - 1) = 0$