Algebra Examples

$\frac{3 a^{3} - 15 a^{2} b + 12 a b^{2}}{3 a^{2} b}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Factor $3 a$ out of $3 a^{3} - 15 a^{2} b + 12 a b^{2}$ .

Tap for more steps...

Step 1.1.1

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

$\frac{3 a (a^{2}) - 15 a^{2} b + 12 a b^{2}}{3 a^{2} b}$

Step 1.1.2

Factor $3 a$ out of $- 15 a^{2} b$ .

$\frac{3 a (a^{2}) + 3 a (- 5 a b) + 12 a b^{2}}{3 a^{2} b}$

Step 1.1.3

Factor $3 a$ out of $12 a b^{2}$ .

$\frac{3 a (a^{2}) + 3 a (- 5 a b) + 3 a (4 b^{2})}{3 a^{2} b}$

Step 1.1.4

Factor $3 a$ out of $3 a (a^{2}) + 3 a (- 5 a b)$ .

$\frac{3 a (a^{2} - 5 a b) + 3 a (4 b^{2})}{3 a^{2} b}$

Step 1.1.5

Factor $3 a$ out of $3 a (a^{2} - 5 a b) + 3 a (4 b^{2})$ .

$\frac{3 a (a^{2} - 5 a b + 4 b^{2})}{3 a^{2} b}$

Step 1.2

Factor by grouping.

Tap for more steps...

Step 1.2.1

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 $a \cdot c = 1 \cdot 4 = 4$ and whose sum is $b = - 5$ .

Tap for more steps...

Step 1.2.1.1

Reorder terms.

$\frac{3 a (a^{2} + 4 b^{2} - 5 a b)}{3 a^{2} b}$

Step 1.2.1.2

Reorder $4 b^{2}$ and $- 5 a b$ .

$\frac{3 a (a^{2} - 5 a b + 4 b^{2})}{3 a^{2} b}$

Step 1.2.1.3

Factor $- 5$ out of $- 5 a b$ .

$\frac{3 a (a^{2} - 5 (a b) + 4 b^{2})}{3 a^{2} b}$

Step 1.2.1.4

Rewrite $- 5$ as $- 1$ plus $- 4$

$\frac{3 a (a^{2} + (- 1 - 4) (a b) + 4 b^{2})}{3 a^{2} b}$

Step 1.2.1.5

Apply the distributive property.

$\frac{3 a (a^{2} - 1 (a b) - 4 (a b) + 4 b^{2})}{3 a^{2} b}$

Step 1.2.1.6

Move parentheses.

$\frac{3 a (a^{2} - 1 a b - 4 a b + 4 b^{2})}{3 a^{2} b}$

Step 1.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{3 a ((a^{2} - 1 a b) - 4 a b + 4 b^{2})}{3 a^{2} b}$

Step 1.2.2.2

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

$\frac{3 a (a (a - 1 b) - 4 b (a - b))}{3 a^{2} b}$

Step 1.2.3

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

$\frac{3 a ((a - 1 b) (a - 4 b))}{3 a^{2} b}$

Step 1.3

Rewrite $- 1 b$ as $- b$ .

$\frac{3 a (a - b) (a - 4 b)}{3 a^{2} b}$

Step 2

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.1.1

Cancel the common factor.

$\frac{3 a (a - b) (a - 4 b)}{3 a^{2} b}$

Step 2.1.2

Rewrite the expression.

$\frac{(a (a - b)) (a - 4 b)}{a^{2} b}$

Step 2.2

Cancel the common factor of $a$ and $a^{2}$ .

Tap for more steps...

Step 2.2.1

Factor $a$ out of $(a (a - b)) (a - 4 b)$ .

$\frac{a ((a - b) (a - 4 b))}{a^{2} b}$

Step 2.2.2

Cancel the common factors.

Tap for more steps...

Step 2.2.2.1

Factor $a$ out of $a^{2} b$ .

$\frac{a ((a - b) (a - 4 b))}{a (a b)}$

Step 2.2.2.2

Cancel the common factor.

$\frac{a ((a - b) (a - 4 b))}{a (a b)}$

Step 2.2.2.3

Rewrite the expression.

$\frac{(a - b) (a - 4 b)}{a b}$