Divide ((x^2-4)/(x^3+7x^2))÷((x^3-x^2-6x)/(x^2+4x-21))

$\frac{x^{2} - 4}{x^{3} + 7 x^{2}} \div \frac{x^{3} - x^{2} - 6 x}{x^{2} + 4 x - 21}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 4}{x^{3} + 7 x^{2}} \cdot \frac{x^{2} + 4 x - 21}{x^{3} - x^{2} - 6 x}$

Step 2

Simplify the numerator.

Rewrite $4$ as $2^{2}$ .

$\frac{x^{2} - 2^{2}}{x^{3} + 7 x^{2}} \cdot \frac{x^{2} + 4 x - 21}{x^{3} - x^{2} - 6 x}$

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 $b = 2$ .

$\frac{(x + 2) (x - 2)}{x^{3} + 7 x^{2}} \cdot \frac{x^{2} + 4 x - 21}{x^{3} - x^{2} - 6 x}$

Step 3

Factor $x^{2}$ out of $x^{3} + 7 x^{2}$ .

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

$\frac{(x + 2) (x - 2)}{x^{2} x + 7 x^{2}} \cdot \frac{x^{2} + 4 x - 21}{x^{3} - x^{2} - 6 x}$

Factor $x^{2}$ out of $7 x^{2}$ .

$\frac{(x + 2) (x - 2)}{x^{2} x + x^{2} \cdot 7} \cdot \frac{x^{2} + 4 x - 21}{x^{3} - x^{2} - 6 x}$

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 7$ .

$\frac{(x + 2) (x - 2)}{x^{2} (x + 7)} \cdot \frac{x^{2} + 4 x - 21}{x^{3} - x^{2} - 6 x}$

Step 4

Factor $x^{2} + 4 x - 21$ 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 $- 21$ and whose sum is $4$ .

$- 3, 7$

Write the factored form using these integers.

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

Step 5

Simplify the denominator.

Factor $x$ out of $x^{3} - x^{2} - 6 x$ .

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

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

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

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

Factor $x$ out of $- 6 x$ .

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

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

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

Factor $x$ out of $x (x^{2} - x) + x \cdot - 6$ .

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

Factor $x^{2} - x - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Write the factored form using these integers.

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

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

Step 6

Combine.

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

Step 7

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x^{2}$ .

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

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $2$ .

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

Step 8

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 9

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 10

Cancel the common factor of $x + 7$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 11

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

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

Step 12

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

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

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 13

Move the negative in front of the fraction.

$\frac{1}{x^{2}} - \frac{2}{x^{3}}$

Divide ((x^2-4)/(x^3+7x^2))÷((x^3-x^2-6x)/(x^2+4x-21))

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