Precalculus Examples

$\frac{x^{2} - 13 x + 40}{x^{4} - 25 x^{2}}$

Step 1

Factor $x^{2} - 13 x + 40$ using the AC method.

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Step 1.1

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 $40$ and whose sum is $- 13$ .

$- 8, - 5$

Step 1.2

Write the factored form using these integers.

$\frac{(x - 8) (x - 5)}{x^{4} - 25 x^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $x^{2}$ out of $x^{4} - 25 x^{2}$ .

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Step 2.1.1

Factor $x^{2}$ out of $x^{4}$ .

$\frac{(x - 8) (x - 5)}{x^{2} x^{2} - 25 x^{2}}$

Step 2.1.2

Factor $x^{2}$ out of $- 25 x^{2}$ .

$\frac{(x - 8) (x - 5)}{x^{2} x^{2} + x^{2} \cdot - 25}$

Step 2.1.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 25$ .

$\frac{(x - 8) (x - 5)}{x^{2} (x^{2} - 25)}$

Step 2.2

Rewrite $25$ as $5^{2}$ .

$\frac{(x - 8) (x - 5)}{x^{2} (x^{2} - 5^{2})}$

Step 2.3

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 $b = 5$ .

$\frac{(x - 8) (x - 5)}{x^{2} (x + 5) (x - 5)}$

Step 3

Cancel the common factor of $x - 5$ .

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Step 3.1

Cancel the common factor.

$\frac{(x - 8) (x - 5)}{x^{2} (x + 5) (x - 5)}$

Step 3.2

Rewrite the expression.

$\frac{x - 8}{x^{2} (x + 5)}$