Basic Math Examples

$\frac{k - 2}{k^{2} - 2 k - 12} - 3 k$

Step 1

To write $- 3 k$ as a fraction with a common denominator, multiply by $\frac{k^{2} - 2 k - 12}{k^{2} - 2 k - 12}$ .

$\frac{k - 2}{k^{2} - 2 k - 12} - 3 k \cdot \frac{k^{2} - 2 k - 12}{k^{2} - 2 k - 12}$

Step 2

Simplify terms.

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Step 2.1

Combine $- 3 k$ and $\frac{k^{2} - 2 k - 12}{k^{2} - 2 k - 12}$ .

$\frac{k - 2}{k^{2} - 2 k - 12} + \frac{- 3 k (k^{2} - 2 k - 12)}{k^{2} - 2 k - 12}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{k - 2 - 3 k (k^{2} - 2 k - 12)}{k^{2} - 2 k - 12}$

Step 3

Simplify the numerator.

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Step 3.1

Apply the distributive property.

$\frac{k - 2 - 3 k \cdot k^{2} - 3 k (- 2 k) - 3 k \cdot - 12}{k^{2} - 2 k - 12}$

Step 3.2

Simplify.

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Step 3.2.1

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

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Step 3.2.1.1

Move $k^{2}$ .

$\frac{k - 2 - 3 (k^{2} k) - 3 k (- 2 k) - 3 k \cdot - 12}{k^{2} - 2 k - 12}$

Step 3.2.1.2

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

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Step 3.2.1.2.1

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

$\frac{k - 2 - 3 (k^{2} k^{1}) - 3 k (- 2 k) - 3 k \cdot - 12}{k^{2} - 2 k - 12}$

Step 3.2.1.2.2

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

$\frac{k - 2 - 3 k^{2 + 1} - 3 k (- 2 k) - 3 k \cdot - 12}{k^{2} - 2 k - 12}$

Step 3.2.1.3

Add $2$ and $1$ .

$\frac{k - 2 - 3 k^{3} - 3 k (- 2 k) - 3 k \cdot - 12}{k^{2} - 2 k - 12}$

Step 3.2.2

Rewrite using the commutative property of multiplication.

$\frac{k - 2 - 3 k^{3} - 3 \cdot - 2 k \cdot k - 3 k \cdot - 12}{k^{2} - 2 k - 12}$

Step 3.2.3

Multiply $- 12$ by $- 3$ .

$\frac{k - 2 - 3 k^{3} - 3 \cdot - 2 k \cdot k + 36 k}{k^{2} - 2 k - 12}$

Step 3.3

Simplify each term.

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Step 3.3.1

Multiply $k$ by $k$ by adding the exponents.

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Step 3.3.1.1

Move $k$ .

$\frac{k - 2 - 3 k^{3} - 3 \cdot - 2 (k \cdot k) + 36 k}{k^{2} - 2 k - 12}$

Step 3.3.1.2

Multiply $k$ by $k$ .

$\frac{k - 2 - 3 k^{3} - 3 \cdot - 2 k^{2} + 36 k}{k^{2} - 2 k - 12}$

Step 3.3.2

Multiply $- 3$ by $- 2$ .

$\frac{k - 2 - 3 k^{3} + 6 k^{2} + 36 k}{k^{2} - 2 k - 12}$

Step 3.4

Add $k$ and $36 k$ .

$\frac{- 2 - 3 k^{3} + 6 k^{2} + 37 k}{k^{2} - 2 k - 12}$

Step 3.5

Reorder terms.

$\frac{- 3 k^{3} + 6 k^{2} + 37 k - 2}{k^{2} - 2 k - 12}$

Step 4

Simplify with factoring out.

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Step 4.1

Factor $- 1$ out of $- 3 k^{3}$ .

$\frac{- (3 k^{3}) + 6 k^{2} + 37 k - 2}{k^{2} - 2 k - 12}$

Step 4.2

Factor $- 1$ out of $6 k^{2}$ .

$\frac{- (3 k^{3}) - (- 6 k^{2}) + 37 k - 2}{k^{2} - 2 k - 12}$

Step 4.3

Factor $- 1$ out of $- (3 k^{3}) - (- 6 k^{2})$ .

$\frac{- (3 k^{3} - 6 k^{2}) + 37 k - 2}{k^{2} - 2 k - 12}$

Step 4.4

Factor $- 1$ out of $37 k$ .

$\frac{- (3 k^{3} - 6 k^{2}) - (- 37 k) - 2}{k^{2} - 2 k - 12}$

Step 4.5

Factor $- 1$ out of $- (3 k^{3} - 6 k^{2}) - (- 37 k)$ .

$\frac{- (3 k^{3} - 6 k^{2} - 37 k) - 2}{k^{2} - 2 k - 12}$

Step 4.6

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- (3 k^{3} - 6 k^{2} - 37 k) - 1 (2)}{k^{2} - 2 k - 12}$

Step 4.7

Factor $- 1$ out of $- (3 k^{3} - 6 k^{2} - 37 k) - 1 (2)$ .

$\frac{- (3 k^{3} - 6 k^{2} - 37 k + 2)}{k^{2} - 2 k - 12}$

Step 4.8

Simplify the expression.

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Step 4.8.1

Rewrite $- (3 k^{3} - 6 k^{2} - 37 k + 2)$ as $- 1 (3 k^{3} - 6 k^{2} - 37 k + 2)$ .

$\frac{- 1 (3 k^{3} - 6 k^{2} - 37 k + 2)}{k^{2} - 2 k - 12}$

Step 4.8.2

Move the negative in front of the fraction.

$- \frac{3 k^{3} - 6 k^{2} - 37 k + 2}{k^{2} - 2 k - 12}$