Finite Math Examples

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

Step 1

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

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Step 1.1

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

$\frac{3 a b (5 a^{2}) - 12 a^{2} b^{2} + 15 a b^{3}}{- 3 a}$

Step 1.2

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

$\frac{3 a b (5 a^{2}) + 3 a b (- 4 a b) + 15 a b^{3}}{- 3 a}$

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

Cancel the common factor of $3$ and $- 3$ .

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Step 2.1

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

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

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $3$ out of $- 3 a$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

Cancel the common factor of $a$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

$\frac{b (5 a^{2} - 4 a b + 5 b^{2})}{- 1}$

Step 3.3

Move the negative one from the denominator of $\frac{b (5 a^{2} - 4 a b + 5 b^{2})}{- 1}$ .

$- 1 \cdot (b (5 a^{2} - 4 a b + 5 b^{2}))$

Step 4

Rewrite $- 1 \cdot (b (5 a^{2} - 4 a b + 5 b^{2}))$ as $- (b (5 a^{2} - 4 a b + 5 b^{2}))$ .

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

Step 5

Apply the distributive property.

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

Step 6

Simplify.

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Step 6.1

Rewrite using the commutative property of multiplication.

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

Step 6.2

Rewrite using the commutative property of multiplication.

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

Step 6.3

Rewrite using the commutative property of multiplication.

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

Step 7

Simplify each term.

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Step 7.1

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

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Step 7.1.1

Move $b$ .

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

Step 7.1.2

Multiply $b$ by $b$ .

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

Step 7.2

Multiply $b$ by $b^{2}$ by adding the exponents.

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Step 7.2.1

Move $b^{2}$ .

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

Step 7.2.2

Multiply $b^{2}$ by $b$ .

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Step 7.2.2.1

Raise $b$ to the power of $1$ .

$- (5 b a^{2} - 4 b^{2} a + 5 (b^{2} b^{1}))$

Step 7.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- (5 b a^{2} - 4 b^{2} a + 5 b^{2 + 1})$

Step 7.2.3

Add $2$ and $1$ .

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

Step 8

Apply the distributive property.

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

Step 9

Simplify.

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Step 9.1

Multiply $5$ by $- 1$ .

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

Step 9.2

Multiply $- 4$ by $- 1$ .

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

Step 9.3

Multiply $5$ by $- 1$ .

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

Step 10

Remove parentheses.

$- 5 b a^{2} + 4 b^{2} a - 5 b^{3}$