Finite Math Examples

$\frac{x^{2} - 16 x}{x^{2} - x - 16} \div \frac{x^{3} - x^{2} - 20 x}{x^{2} - 8 x + 15}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 16 x}{x^{2} - x - 16} \cdot \frac{x^{2} - 8 x + 15}{x^{3} - x^{2} - 20 x}$

Step 2

Factor $x$ out of $x^{2} - 16 x$ .

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Step 2.1

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

$\frac{x \cdot x - 16 x}{x^{2} - x - 16} \cdot \frac{x^{2} - 8 x + 15}{x^{3} - x^{2} - 20 x}$

Step 2.2

Factor $x$ out of $- 16 x$ .

$\frac{x \cdot x + x \cdot - 16}{x^{2} - x - 16} \cdot \frac{x^{2} - 8 x + 15}{x^{3} - x^{2} - 20 x}$

Step 2.3

Factor $x$ out of $x \cdot x + x \cdot - 16$ .

$\frac{x (x - 16)}{x^{2} - x - 16} \cdot \frac{x^{2} - 8 x + 15}{x^{3} - x^{2} - 20 x}$

Step 3

Factor $x^{2} - 8 x + 15$ 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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 3.2

Write the factored form using these integers.

$\frac{x (x - 16)}{x^{2} - x - 16} \cdot \frac{(x - 5) (x - 3)}{x^{3} - x^{2} - 20 x}$

Step 4

Simplify the denominator.

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Step 4.1

Factor $x$ out of $x^{3} - x^{2} - 20 x$ .

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Step 4.1.1

Factor $x$ out of $x^{3}$ .

$\frac{x (x - 16)}{x^{2} - x - 16} \cdot \frac{(x - 5) (x - 3)}{x \cdot x^{2} - x^{2} - 20 x}$

Step 4.1.2

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

$\frac{x (x - 16)}{x^{2} - x - 16} \cdot \frac{(x - 5) (x - 3)}{x \cdot x^{2} + x (- x) - 20 x}$

Step 4.1.3

Factor $x$ out of $- 20 x$ .

$\frac{x (x - 16)}{x^{2} - x - 16} \cdot \frac{(x - 5) (x - 3)}{x \cdot x^{2} + x (- x) + x \cdot - 20}$

Step 4.1.4

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

$\frac{x (x - 16)}{x^{2} - x - 16} \cdot \frac{(x - 5) (x - 3)}{x (x^{2} - x) + x \cdot - 20}$

Step 4.1.5

Factor $x$ out of $x (x^{2} - x) + x \cdot - 20$ .

$\frac{x (x - 16)}{x^{2} - x - 16} \cdot \frac{(x - 5) (x - 3)}{x (x^{2} - x - 20)}$

Step 4.2

Factor $x^{2} - x - 20$ using the AC method.

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Step 4.2.1

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 $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Step 4.2.2

Write the factored form using these integers.

$\frac{x (x - 16)}{x^{2} - x - 16} \cdot \frac{(x - 5) (x - 3)}{x ((x - 5) (x + 4))}$

$\frac{x (x - 16)}{x^{2} - x - 16} \cdot \frac{(x - 5) (x - 3)}{x (x - 5) (x + 4)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x$ .

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Step 5.1.1

Factor $x$ out of $x (x - 5) (x + 4)$ .

$\frac{x (x - 16)}{x^{2} - x - 16} \cdot \frac{(x - 5) (x - 3)}{x ((x - 5) (x + 4))}$

Step 5.1.2

Cancel the common factor.

$\frac{x (x - 16)}{x^{2} - x - 16} \cdot \frac{(x - 5) (x - 3)}{x ((x - 5) (x + 4))}$

Step 5.1.3

Rewrite the expression.

$\frac{x - 16}{x^{2} - x - 16} \cdot \frac{(x - 5) (x - 3)}{(x - 5) (x + 4)}$

Step 5.2

Multiply $\frac{x - 16}{x^{2} - x - 16}$ by $\frac{(x - 5) (x - 3)}{(x - 5) (x + 4)}$ .

$\frac{(x - 16) ((x - 5) (x - 3))}{(x^{2} - x - 16) ((x - 5) (x + 4))}$

Step 5.3

Cancel the common factor of $x - 5$ .

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Step 5.3.1

Cancel the common factor.

$\frac{(x - 16) ((x - 5) (x - 3))}{(x^{2} - x - 16) ((x - 5) (x + 4))}$

Step 5.3.2

Rewrite the expression.

$\frac{(x - 16) (x - 3)}{(x^{2} - x - 16) (x + 4)}$