Simplify ((x^2-2x-15)/(x-3))÷((x-5)/(x^2-9))

$\frac{x^{2} - 2 x - 15}{x - 3} \div \frac{x - 5}{x^{2} - 9}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 2 x - 15}{x - 3} \cdot \frac{x^{2} - 9}{x - 5}$

Step 2

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

$- 5, 3$

Write the factored form using these integers.

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

Step 3

Simplify terms.

Cancel the common factor of $x - 5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{x + 3}{x - 3}$ by $x^{2} - 9$ .

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

Step 4

Simplify the numerator.

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

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\frac{(x + 3) (x + 3) (x - 3)}{x - 3}$

Combine exponents.

Raise $x + 3$ to the power of $1$ .

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

Raise $x + 3$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Step 5

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Divide ${(x + 3)}^{2}$ by $1$ .

${(x + 3)}^{2}$

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$(x + 3) (x + 3)$

Step 6

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Apply the distributive property.

$x (x + 3) + 3 (x + 3)$

Apply the distributive property.

$x \cdot x + x \cdot 3 + 3 (x + 3)$

Apply the distributive property.

$x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3$

Step 7

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot 3 + 3 x + 3 \cdot 3$

Move $3$ to the left of $x$ .

$x^{2} + 3 \cdot x + 3 x + 3 \cdot 3$

Multiply $3$ by $3$ .

$x^{2} + 3 x + 3 x + 9$

Add $3 x$ and $3 x$ .

$x^{2} + 6 x + 9$

Simplify ((x^2-2x-15)/(x-3))÷((x-5)/(x^2-9))

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