Algebra Examples

$\frac{m^{2} + m - m n - n}{m^{2} + m + m n + n} \div \frac{m^{2} - m - m n + n}{m^{2} - m + m n - n}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{m^{2} + m - m n - n}{m^{2} + m + m n + n} \cdot \frac{m^{2} - m + m n - n}{m^{2} - m - m n + n}$

Step 2

Simplify the numerator.

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Step 2.1

Factor out the greatest common factor from each group.

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Step 2.1.1

Group the first two terms and the last two terms.

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

Step 2.1.2

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

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

Step 2.2

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

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

Step 3

Simplify the denominator.

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Step 3.1

Factor out the greatest common factor from each group.

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Step 3.1.1

Group the first two terms and the last two terms.

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

Step 3.1.2

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

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

Step 3.2

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

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

Step 4

Cancel the common factor of $m + 1$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

$\frac{m - n}{m + n} \cdot \frac{m^{2} - m + m n - n}{m^{2} - m - m n + n}$

Step 5

Simplify the numerator.

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Step 5.1

Factor out the greatest common factor from each group.

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Step 5.1.1

Group the first two terms and the last two terms.

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

Step 5.1.2

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

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

Step 5.2

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

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

Step 6

Simplify the denominator.

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Step 6.1

Factor out the greatest common factor from each group.

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Step 6.1.1

Group the first two terms and the last two terms.

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

Step 6.1.2

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

$\frac{m - n}{m + n} \cdot \frac{(m - 1) (m + n)}{m (m - 1) - n (m - 1)}$

Step 6.2

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

$\frac{m - n}{m + n} \cdot \frac{(m - 1) (m + n)}{(m - 1) (m - n)}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Cancel the common factor of $m - n$ .

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Step 7.1.1

Factor $m - n$ out of $(m - 1) (m - n)$ .

$\frac{m - n}{m + n} \cdot \frac{(m - 1) (m + n)}{(m - n) (m - 1)}$

Step 7.1.2

Cancel the common factor.

$\frac{m - n}{m + n} \cdot \frac{(m - 1) (m + n)}{(m - n) (m - 1)}$

Step 7.1.3

Rewrite the expression.

$\frac{1}{m + n} \cdot \frac{(m - 1) (m + n)}{m - 1}$

Step 7.2

Cancel the common factor of $m + n$ .

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Step 7.2.1

Factor $m + n$ out of $(m - 1) (m + n)$ .

$\frac{1}{m + n} \cdot \frac{(m + n) (m - 1)}{m - 1}$

Step 7.2.2

Cancel the common factor.

$\frac{1}{m + n} \cdot \frac{(m + n) (m - 1)}{m - 1}$

Step 7.2.3

Rewrite the expression.

$\frac{m - 1}{m - 1}$

Step 7.3

Cancel the common factor of $m - 1$ .

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Step 7.3.1

Cancel the common factor.

$\frac{m - 1}{m - 1}$

Step 7.3.2

Rewrite the expression.

$1$