Finite Math Examples

$\frac{x^{4} - 8 x^{2} + 15}{x^{4} - 9}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(x^{2})}^{2} - 8 x^{2} + 15}{x^{4} - 9}$

Step 1.2

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

$\frac{u^{2} - 8 u + 15}{x^{4} - 9}$

Step 1.3

Factor $u^{2} - 8 u + 15$ using the AC method.

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Step 1.3.1

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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 1.3.2

Write the factored form using these integers.

$\frac{(u - 5) (u - 3)}{x^{4} - 9}$

Step 1.4

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

$\frac{(x^{2} - 5) (x^{2} - 3)}{x^{4} - 9}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{(x^{2} - 5) (x^{2} - 3)}{{(x^{2})}^{2} - 9}$

Step 2.2

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

$\frac{(x^{2} - 5) (x^{2} - 3)}{{(x^{2})}^{2} - 3^{2}}$

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 $a = x^{2}$ and $b = 3$ .

$\frac{(x^{2} - 5) (x^{2} - 3)}{(x^{2} + 3) (x^{2} - 3)}$

Step 3

Cancel the common factor of $x^{2} - 3$ .

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Step 3.1

Cancel the common factor.

$\frac{(x^{2} - 5) (x^{2} - 3)}{(x^{2} + 3) (x^{2} - 3)}$

Step 3.2

Rewrite the expression.

$\frac{x^{2} - 5}{x^{2} + 3}$