Pre-Algebra Examples

$\frac{k + 5}{k + 1} + \frac{8}{k^{2} - 1}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

$\frac{k + 5}{k + 1} + \frac{8}{k^{2} - 1^{2}}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = k$ and $b = 1$ .

$\frac{k + 5}{k + 1} + \frac{8}{(k + 1) (k - 1)}$

Step 2

To write $\frac{k + 5}{k + 1}$ as a fraction with a common denominator, multiply by $\frac{k - 1}{k - 1}$ .

$\frac{k + 5}{k + 1} \cdot \frac{k - 1}{k - 1} + \frac{8}{(k + 1) (k - 1)}$

Step 3

Simplify terms.

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Step 3.1

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

$\frac{(k + 5) (k - 1)}{(k + 1) (k - 1)} + \frac{8}{(k + 1) (k - 1)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{(k + 5) (k - 1) + 8}{(k + 1) (k - 1)}$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(k + 5) (k - 1)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{k (k - 1) + 5 (k - 1) + 8}{(k + 1) (k - 1)}$

Step 4.1.2

Apply the distributive property.

$\frac{k \cdot k + k \cdot - 1 + 5 (k - 1) + 8}{(k + 1) (k - 1)}$

Step 4.1.3

Apply the distributive property.

$\frac{k \cdot k + k \cdot - 1 + 5 k + 5 \cdot - 1 + 8}{(k + 1) (k - 1)}$

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $k$ by $k$ .

$\frac{k^{2} + k \cdot - 1 + 5 k + 5 \cdot - 1 + 8}{(k + 1) (k - 1)}$

Step 4.2.1.2

Move $- 1$ to the left of $k$ .

$\frac{k^{2} - 1 \cdot k + 5 k + 5 \cdot - 1 + 8}{(k + 1) (k - 1)}$

Step 4.2.1.3

Rewrite $- 1 k$ as $- k$ .

$\frac{k^{2} - k + 5 k + 5 \cdot - 1 + 8}{(k + 1) (k - 1)}$

Step 4.2.1.4

Multiply $5$ by $- 1$ .

$\frac{k^{2} - k + 5 k - 5 + 8}{(k + 1) (k - 1)}$

Step 4.2.2

Add $- k$ and $5 k$ .

$\frac{k^{2} + 4 k - 5 + 8}{(k + 1) (k - 1)}$

Step 4.3

Add $- 5$ and $8$ .

$\frac{k^{2} + 4 k + 3}{(k + 1) (k - 1)}$

Step 4.4

Factor $k^{2} + 4 k + 3$ using the AC method.

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Step 4.4.1

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 $4$ .

$1, 3$

Step 4.4.2

Write the factored form using these integers.

$\frac{(k + 1) (k + 3)}{(k + 1) (k - 1)}$

Step 5

Cancel the common factor of $k + 1$ .

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Step 5.1

Cancel the common factor.

$\frac{(k + 1) (k + 3)}{(k + 1) (k - 1)}$

Step 5.2

Rewrite the expression.

$\frac{k + 3}{k - 1}$