Algebra Examples

$\frac{x^{2} - 4 x - 5}{x + 5} \div (x^{2} + 6 x + 5)$

Step 1

Rewrite the division as a fraction.

$\frac{\frac{x^{2} - 4 x - 5}{x + 5}}{x^{2} + 6 x + 5}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 4 x - 5}{x + 5} \cdot \frac{1}{x^{2} + 6 x + 5}$

Step 3

Factor $x^{2} - 4 x - 5$ 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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 5) (x + 1)}{x + 5} \cdot \frac{1}{x^{2} + 6 x + 5}$

Step 4

Factor $x^{2} + 6 x + 5$ 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 $5$ and whose sum is $6$ .

$1, 5$

Step 4.2

Write the factored form using these integers.

$\frac{(x - 5) (x + 1)}{x + 5} \cdot \frac{1}{(x + 1) (x + 5)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x + 1$ .

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Step 5.1.1

Factor $x + 1$ out of $(x - 5) (x + 1)$ .

$\frac{(x + 1) (x - 5)}{x + 5} \cdot \frac{1}{(x + 1) (x + 5)}$

Step 5.1.2

Cancel the common factor.

$\frac{(x + 1) (x - 5)}{x + 5} \cdot \frac{1}{(x + 1) (x + 5)}$

Step 5.1.3

Rewrite the expression.

$\frac{x - 5}{x + 5} \cdot \frac{1}{x + 5}$

Step 5.2

Multiply $\frac{x - 5}{x + 5}$ by $\frac{1}{x + 5}$ .

$\frac{x - 5}{(x + 5) (x + 5)}$

Step 6

Raise $x + 5$ to the power of $1$ .

$\frac{x - 5}{{(x + 5)}^{1} (x + 5)}$

Step 7

Raise $x + 5$ to the power of $1$ .

$\frac{x - 5}{{(x + 5)}^{1} {(x + 5)}^{1}}$

Step 8

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

$\frac{x - 5}{{(x + 5)}^{1 + 1}}$

Step 9

Add $1$ and $1$ .

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