Pre-Algebra Examples

$\frac{x}{x^{4} - 5 x^{2} + 4}$

Step 1

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

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

Step 2

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

$\frac{x}{u^{2} - 5 u + 4}$

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 $- 5$ .

$- 4, - 1$

Step 3.2

Write the factored form using these integers.

$\frac{x}{(u - 4) (u - 1)}$

Step 4

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

$\frac{x}{(x^{2} - 4) (x^{2} - 1)}$

Step 5

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

$\frac{x}{(x^{2} - 2^{2}) (x^{2} - 1)}$

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 $b = 2$ .

$\frac{x}{(x + 2) (x - 2) (x^{2} - 1)}$

Step 7

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

$\frac{x}{(x + 2) (x - 2) (x^{2} - 1^{2})}$

Step 8

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 $b = 1$ .

$\frac{x}{(x + 2) (x - 2) (x + 1) (x - 1)}$