Precalculus Examples

$\frac{5 + \frac{1}{m} - \frac{6}{m^{2}}}{\frac{2}{m} - \frac{2}{m^{2}}}$

Step 1

Multiply the numerator and denominator of the fraction by $m^{2}$ .

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Step 1.1

Multiply $\frac{5 + \frac{1}{m} - \frac{6}{m^{2}}}{\frac{2}{m} - \frac{2}{m^{2}}}$ by $\frac{m^{2}}{m^{2}}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $m$ .

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Step 3.1.1

Factor $m$ out of $m^{2}$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $m^{2}$ .

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Step 3.2.1

Move the leading negative in $- \frac{6}{m^{2}}$ into the numerator.

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $m$ .

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Step 3.3.1

Factor $m$ out of $m^{2}$ .

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $m^{2}$ .

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Step 3.4.1

Move the leading negative in $- \frac{2}{m^{2}}$ into the numerator.

$\frac{m^{2} \cdot 5 + m - 6}{m \cdot 2 + m^{2} \frac{- 2}{m^{2}}}$

Step 3.4.2

Cancel the common factor.

$\frac{m^{2} \cdot 5 + m - 6}{m \cdot 2 + m^{2} \frac{- 2}{m^{2}}}$

Step 3.4.3

Rewrite the expression.

$\frac{m^{2} \cdot 5 + m - 6}{m \cdot 2 - 2}$

Step 4

Factor by grouping.

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Step 4.1

Reorder terms.

$\frac{5 m^{2} + m - 6}{m \cdot 2 - 2}$

Step 4.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 6 = - 30$ and whose sum is $b = 1$ .

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Step 4.2.1

Multiply by $1$ .

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

Step 4.2.2

Rewrite $1$ as $- 5$ plus $6$

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

Step 4.2.3

Apply the distributive property.

$\frac{5 m^{2} - 5 m + 6 m - 6}{m \cdot 2 - 2}$

Step 4.3

Factor out the greatest common factor from each group.

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Step 4.3.1

Group the first two terms and the last two terms.

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

Step 4.3.2

Factor out the greatest common factor (GCF) from each group.

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

Step 4.4

Factor the polynomial by factoring out the greatest common factor, $m - 1$ .

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

Step 5

Factor $2$ out of $m \cdot 2 - 2$ .

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Step 5.1

Factor $2$ out of $m \cdot 2$ .

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

Step 5.2

Factor $2$ out of $- 2$ .

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

Step 5.3

Factor $2$ out of $2 \cdot m + 2 (- 1)$ .

$\frac{(m - 1) (5 m + 6)}{2 (m - 1)}$

Step 6

Cancel the common factor of $m - 1$ .

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Step 6.1

Cancel the common factor.

$\frac{(m - 1) (5 m + 6)}{2 (m - 1)}$

Step 6.2

Rewrite the expression.

$\frac{5 m + 6}{2}$