Simplify ((x^2-2x-3)/(x^3))÷((x^2-6x-7)/x)

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

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

Simplify terms.

Cancel the common factor of $x$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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 3

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

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 $- 2$ .

$- 3, 1$

Write the factored form using these integers.

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

Step 4

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

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 $- 6$ .

$- 7, 1$

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 5

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify ((x^2-2x-3)/(x^3))÷((x^2-6x-7)/x)

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