Algebra Examples

$x^{4} - 10 x^{2} + 9$

Step 1

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

${(x^{2})}^{2} - 10 x^{2} + 9$

Step 2

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

$u^{2} - 10 u + 9$

Step 3

Factor $u^{2} - 10 u + 9$ 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 $9$ and whose sum is $- 10$ .

$- 9, - 1$

Step 3.2

Write the factored form using these integers.

$(u - 9) (u - 1)$

Step 4

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

$(x^{2} - 9) (x^{2} - 1)$

Step 5

Rewrite $9$ as $3^{2}$ .

$(x^{2} - 3^{2}) (x^{2} - 1)$

Step 6

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 $b = 3$ .

$(x + 3) (x - 3) (x^{2} - 1)$

Step 7

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

$(x + 3) (x - 3) (x^{2} - 1^{2})$

Step 8

Factor.

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Step 8.1

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 $b = 1$ .

$(x + 3) (x - 3) ((x + 1) (x - 1))$

Step 8.2

Remove unnecessary parentheses.

$(x + 3) (x - 3) (x + 1) (x - 1)$