Algebra Examples

$\frac{15 m^{2} n - 3 m n^{2}}{15 m^{2} + 57 m n - 12 n^{2}}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Factor $3 m n$ out of $15 m^{2} n - 3 m n^{2}$ .

Tap for more steps...

Step 1.1.1

Factor $3 m n$ out of $15 m^{2} n$ .

$\frac{3 m n (5 m) - 3 m n^{2}}{15 m^{2} + 57 m n - 12 n^{2}}$

Step 1.1.2

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

$\frac{3 m n (5 m) + 3 m n (- 1 n)}{15 m^{2} + 57 m n - 12 n^{2}}$

Step 1.1.3

Factor $3 m n$ out of $3 m n (5 m) + 3 m n (- 1 n)$ .

$\frac{3 m n (5 m - 1 n)}{15 m^{2} + 57 m n - 12 n^{2}}$

Step 1.2

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

$\frac{3 m n (5 m - n)}{15 m^{2} + 57 m n - 12 n^{2}}$

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

Factor $3$ out of $15 m^{2} + 57 m n - 12 n^{2}$ .

Tap for more steps...

Step 2.1.1

Factor $3$ out of $15 m^{2}$ .

$\frac{3 m n (5 m - n)}{3 (5 m^{2}) + 57 m n - 12 n^{2}}$

Step 2.1.2

Factor $3$ out of $57 m n$ .

$\frac{3 m n (5 m - n)}{3 (5 m^{2}) + 3 (19 m n) - 12 n^{2}}$

Step 2.1.3

Factor $3$ out of $- 12 n^{2}$ .

$\frac{3 m n (5 m - n)}{3 (5 m^{2}) + 3 (19 m n) + 3 (- 4 n^{2})}$

Step 2.1.4

Factor $3$ out of $3 (5 m^{2}) + 3 (19 m n)$ .

$\frac{3 m n (5 m - n)}{3 (5 m^{2} + 19 m n) + 3 (- 4 n^{2})}$

Step 2.1.5

Factor $3$ out of $3 (5 m^{2} + 19 m n) + 3 (- 4 n^{2})$ .

$\frac{3 m n (5 m - n)}{3 (5 m^{2} + 19 m n - 4 n^{2})}$

Step 2.2

Factor by grouping.

Tap for more steps...

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 $a \cdot c = 5 \cdot - 4 = - 20$ and whose sum is $b = 19$ .

Tap for more steps...

Step 2.2.1.1

Reorder terms.

$\frac{3 m n (5 m - n)}{3 (5 m^{2} - 4 n^{2} + 19 m n)}$

Step 2.2.1.2

Reorder $- 4 n^{2}$ and $19 m n$ .

$\frac{3 m n (5 m - n)}{3 (5 m^{2} + 19 m n - 4 n^{2})}$

Step 2.2.1.3

Factor $19$ out of $19 m n$ .

$\frac{3 m n (5 m - n)}{3 (5 m^{2} + 19 (m n) - 4 n^{2})}$

Step 2.2.1.4

Rewrite $19$ as $- 1$ plus $20$

$\frac{3 m n (5 m - n)}{3 (5 m^{2} + (- 1 + 20) (m n) - 4 n^{2})}$

Step 2.2.1.5

Apply the distributive property.

$\frac{3 m n (5 m - n)}{3 (5 m^{2} - 1 (m n) + 20 (m n) - 4 n^{2})}$

Step 2.2.1.6

Move parentheses.

$\frac{3 m n (5 m - n)}{3 (5 m^{2} - 1 m n + 20 m n - 4 n^{2})}$

Step 2.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.2.2.1

Group the first two terms and the last two terms.

$\frac{3 m n (5 m - n)}{3 ((5 m^{2} - 1 m n) + 20 m n - 4 n^{2})}$

Step 2.2.2.2

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

$\frac{3 m n (5 m - n)}{3 (m (5 m - 1 n) + 4 n (5 m - n))}$

Step 2.2.3

Factor the polynomial by factoring out the greatest common factor, $5 m - 1 n$ .

$\frac{3 m n (5 m - n)}{3 ((5 m - 1 n) (m + 4 n))}$

Step 2.3

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

$\frac{3 m n (5 m - n)}{3 (5 m - n) (m + 4 n)}$

Step 3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 3.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.1.1

Cancel the common factor.

$\frac{3 m n (5 m - n)}{3 (5 m - n) (m + 4 n)}$

Step 3.1.2

Rewrite the expression.

$\frac{m n (5 m - n)}{(5 m - n) (m + 4 n)}$

Step 3.2

Cancel the common factor of $5 m - n$ .

Tap for more steps...

Step 3.2.1

Cancel the common factor.

$\frac{m n (5 m - n)}{(5 m - n) (m + 4 n)}$

Step 3.2.2

Rewrite the expression.

$\frac{m n}{m + 4 n}$