Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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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.

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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 $b = - 5$ .

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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.

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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

Cancel the common factor of $3$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Rewrite the expression.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Expand $(a - b) (a - 4 b)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $a$ by $a$ .

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

Step 5.1.2

Rewrite using the commutative property of multiplication.

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

Step 5.1.3

Rewrite using the commutative property of multiplication.

$\frac{a^{2} - 4 a b - b a - 1 \cdot - 4 b \cdot b}{a b}$

Step 5.1.4

Multiply $b$ by $b$ by adding the exponents.

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Step 5.1.4.1

Move $b$ .

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

Step 5.1.4.2

Multiply $b$ by $b$ .

$\frac{a^{2} - 4 a b - b a - 1 \cdot - 4 b^{2}}{a b}$

Step 5.1.5

Multiply $- 1$ by $- 4$ .

$\frac{a^{2} - 4 a b - b a + 4 b^{2}}{a b}$

Step 5.2

Subtract $b a$ from $- 4 a b$ .

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Step 5.2.1

Move $b$ .

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

Step 5.2.2

Subtract $a b$ from $- 4 a b$ .

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

Step 6

Split the fraction $\frac{a^{2} - 5 a b + 4 b^{2}}{a b}$ into two fractions.

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

Step 7

Split the fraction $\frac{a^{2} - 5 a b}{a b}$ into two fractions.

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

Step 8

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

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Step 8.1

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

$\frac{a \cdot a}{a b} + \frac{- 5 a b}{a b} + \frac{4 b^{2}}{a b}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $a$ out of $a b$ .

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

Step 8.2.2

Cancel the common factor.

$\frac{a \cdot a}{a b} + \frac{- 5 a b}{a b} + \frac{4 b^{2}}{a b}$

Step 8.2.3

Rewrite the expression.

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

Step 9

Cancel the common factor of $a$ .

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Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

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

Step 10

Cancel the common factor of $b$ .

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Step 10.1

Cancel the common factor.

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

Step 10.2

Divide $- 5$ by $1$ .

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

Step 11

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

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Step 11.1

Factor $b$ out of $4 b^{2}$ .

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

Step 11.2

Cancel the common factors.

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Step 11.2.1

Factor $b$ out of $a b$ .

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

Step 11.2.2

Cancel the common factor.

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

Step 11.2.3

Rewrite the expression.

$\frac{a}{b} - 5 + \frac{4 b}{a}$