Basic Math Examples

$\frac{m^{3} - m n^{2}}{m n^{2} + m^{2} n - 2 m^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $m$ out of $m^{3} - m n^{2}$ .

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Step 1.1.1

Factor $m$ out of $m^{3}$ .

$\frac{m \cdot m^{2} - m n^{2}}{m n^{2} + m^{2} n - 2 m^{3}}$

Step 1.1.2

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

$\frac{m \cdot m^{2} + m (- 1 n^{2})}{m n^{2} + m^{2} n - 2 m^{3}}$

Step 1.1.3

Factor $m$ out of $m \cdot m^{2} + m (- 1 n^{2})$ .

$\frac{m (m^{2} - 1 n^{2})}{m n^{2} + m^{2} n - 2 m^{3}}$

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 $a = m$ and $b = n$ .

$\frac{m (m + n) (m - n)}{m n^{2} + m^{2} n - 2 m^{3}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $m$ out of $m n^{2} + m^{2} n - 2 m^{3}$ .

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Step 2.1.1

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

$\frac{m (m + n) (m - n)}{m (n^{2}) + m^{2} n - 2 m^{3}}$

Step 2.1.2

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

$\frac{m (m + n) (m - n)}{m (n^{2}) + m (m n) - 2 m^{3}}$

Step 2.1.3

Factor $m$ out of $- 2 m^{3}$ .

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

Step 2.1.4

Factor $m$ out of $m (n^{2}) + m (m n)$ .

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

Step 2.1.5

Factor $m$ out of $m (n^{2} + m n) + m (- 2 m^{2})$ .

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

Step 2.2

Factor by grouping.

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Step 2.2.1

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 $a \cdot c = - 2 \cdot 1 = - 2$ and whose sum is $b = 1$ .

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Step 2.2.1.1

Reorder terms.

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

Step 2.2.1.2

Reorder $n^{2}$ and $m n$ .

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

Step 2.2.1.3

Factor $1$ out of $m n$ .

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

Step 2.2.1.4

Rewrite $1$ as $- 1$ plus $2$

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

Step 2.2.1.5

Apply the distributive property.

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

Step 2.2.1.6

Move parentheses.

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

Step 2.2.2

Factor out the greatest common factor from each group.

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Step 2.2.2.1

Group the first two terms and the last two terms.

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

Step 2.2.2.2

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

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

Step 2.2.3

Factor the polynomial by factoring out the greatest common factor, $- 2 m - 1 n$ .

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

Step 2.3

Rewrite $- 1 n$ as $- n$ .

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $m$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $m - n$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

Factor $- 1$ out of $- 2 m$ .

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

Step 3.4

Factor $- 1$ out of $- n$ .

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

Step 3.5

Factor $- 1$ out of $- (2 m) - (n)$ .

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

Step 3.6

Rewrite negatives.

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Step 3.6.1

Rewrite $- (2 m + n)$ as $- 1 (2 m + n)$ .

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

Step 3.6.2

Move the negative in front of the fraction.

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