Algebra Examples

$\frac{x^{2} - 2 x - 3}{x^{3}} \div \frac{x^{2} - 6 x - 7}{x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 2 x - 3}{x^{3}} \cdot \frac{x}{x^{2} - 6 x - 7}$

Step 2

Cancel the common factor of $x$ .

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Step 2.1

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

$\frac{x^{2} - 2 x - 3}{x \cdot x^{2}} \cdot \frac{x}{x^{2} - 6 x - 7}$

Step 2.2

Cancel the common factor.

$\frac{x^{2} - 2 x - 3}{x \cdot x^{2}} \cdot \frac{x}{x^{2} - 6 x - 7}$

Step 2.3

Rewrite the expression.

$\frac{x^{2} - 2 x - 3}{x^{2}} \cdot \frac{1}{x^{2} - 6 x - 7}$

Step 3

Multiply $\frac{x^{2} - 2 x - 3}{x^{2}}$ by $\frac{1}{x^{2} - 6 x - 7}$ .

$\frac{x^{2} - 2 x - 3}{x^{2} (x^{2} - 6 x - 7)}$

Step 4

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 4.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 4.2

Write the factored form using these integers.

$\frac{(x - 3) (x + 1)}{x^{2} (x^{2} - 6 x - 7)}$

Step 5

Factor $x^{2} - 6 x - 7$ using the AC method.

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Step 5.1

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 $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Step 5.2

Write the factored form using these integers.

$\frac{(x - 3) (x + 1)}{x^{2} ((x - 7) (x + 1))}$

$\frac{(x - 3) (x + 1)}{x^{2} (x - 7) (x + 1)}$

Step 6

Cancel the common factor of $x + 1$ .

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Step 6.1

Cancel the common factor.

$\frac{(x - 3) (x + 1)}{x^{2} (x - 7) (x + 1)}$

Step 6.2

Rewrite the expression.

$\frac{x - 3}{x^{2} (x - 7)}$

Step 7

Split the fraction $\frac{x - 3}{x^{2} (x - 7)}$ into two fractions.

$\frac{x}{x^{2} (x - 7)} + \frac{- 3}{x^{2} (x - 7)}$

Step 8

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

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Step 8.1

Raise $x$ to the power of $1$ .

$\frac{x^{1}}{x^{2} (x - 7)} + \frac{- 3}{x^{2} (x - 7)}$

Step 8.2

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot 1}{x^{2} (x - 7)} + \frac{- 3}{x^{2} (x - 7)}$

Step 8.3

Cancel the common factors.

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Step 8.3.1

Factor $x$ out of $x^{2} (x - 7)$ .

$\frac{x \cdot 1}{x (x (x - 7))} + \frac{- 3}{x^{2} (x - 7)}$

Step 8.3.2

Cancel the common factor.

$\frac{x \cdot 1}{x (x (x - 7))} + \frac{- 3}{x^{2} (x - 7)}$

Step 8.3.3

Rewrite the expression.

$\frac{1}{x (x - 7)} + \frac{- 3}{x^{2} (x - 7)}$

Step 9

Move the negative in front of the fraction.

$\frac{1}{x (x - 7)} - \frac{3}{x^{2} (x - 7)}$