Algebra Examples

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

Step 1

Expand $(2 a - 3 b) (a^{2} - a b + 3 b)$ by multiplying each term in the first expression by each term in the second expression.

$2 a \cdot a^{2} + 2 a (- a b) + 2 a (3 b) - 3 b a^{2} - 3 b (- a b) - 3 b (3 b)$

Step 2

Simplify each term.

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Step 2.1

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

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Step 2.1.1

Move $a^{2}$ .

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

Step 2.1.2

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

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.3

Add $2$ and $1$ .

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

Step 2.2

Multiply $a$ by $a$ by adding the exponents.

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Step 2.2.1

Move $a$ .

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

Step 2.2.2

Multiply $a$ by $a$ .

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

Step 2.3

Rewrite using the commutative property of multiplication.

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

Step 2.4

Multiply $2$ by $- 1$ .

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

Step 2.5

Rewrite using the commutative property of multiplication.

$2 a^{3} - 2 a^{2} b + 2 \cdot 3 a b - 3 b a^{2} - 3 b (- a b) - 3 b (3 b)$

Step 2.6

Multiply $2$ by $3$ .

$2 a^{3} - 2 a^{2} b + 6 a b - 3 b a^{2} - 3 b (- a b) - 3 b (3 b)$

Step 2.7

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

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Step 2.7.1

Move $b$ .

$2 a^{3} - 2 a^{2} b + 6 a b - 3 b a^{2} - 3 (b \cdot b) (- a) - 3 b (3 b)$

Step 2.7.2

Multiply $b$ by $b$ .

$2 a^{3} - 2 a^{2} b + 6 a b - 3 b a^{2} - 3 b^{2} (- a) - 3 b (3 b)$

Step 2.8

Rewrite using the commutative property of multiplication.

$2 a^{3} - 2 a^{2} b + 6 a b - 3 b a^{2} - 3 \cdot - 1 b^{2} a - 3 b (3 b)$

Step 2.9

Multiply $- 3$ by $- 1$ .

$2 a^{3} - 2 a^{2} b + 6 a b - 3 b a^{2} + 3 b^{2} a - 3 b (3 b)$

Step 2.10

Rewrite using the commutative property of multiplication.

$2 a^{3} - 2 a^{2} b + 6 a b - 3 b a^{2} + 3 b^{2} a - 3 \cdot 3 b \cdot b$

Step 2.11

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

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Step 2.11.1

Move $b$ .

$2 a^{3} - 2 a^{2} b + 6 a b - 3 b a^{2} + 3 b^{2} a - 3 \cdot 3 (b \cdot b)$

Step 2.11.2

Multiply $b$ by $b$ .

$2 a^{3} - 2 a^{2} b + 6 a b - 3 b a^{2} + 3 b^{2} a - 3 \cdot 3 b^{2}$

Step 2.12

Multiply $- 3$ by $3$ .

$2 a^{3} - 2 a^{2} b + 6 a b - 3 b a^{2} + 3 b^{2} a - 9 b^{2}$

Step 3

Subtract $3 b a^{2}$ from $- 2 a^{2} b$ .

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Step 3.1

Move $b$ .

$2 a^{3} - 2 a^{2} b - 3 a^{2} b + 6 a b + 3 b^{2} a - 9 b^{2}$

Step 3.2

Subtract $3 a^{2} b$ from $- 2 a^{2} b$ .

$2 a^{3} - 5 a^{2} b + 6 a b + 3 b^{2} a - 9 b^{2}$

Step 4

Move $b^{2}$ .

$2 a^{3} - 5 a^{2} b + 6 a b + 3 a b^{2} - 9 b^{2}$

Step 5

Move $6 a b$ .

$2 a^{3} - 5 a^{2} b + 3 a b^{2} - 9 b^{2} + 6 a b$

Step 6

Move $2 a^{3}$ .

$- 5 a^{2} b + 3 a b^{2} + 2 a^{3} - 9 b^{2} + 6 a b$

Step 7

Reorder $- 5 a^{2} b$ and $3 a b^{2}$ .

$3 a b^{2} - 5 a^{2} b + 2 a^{3} - 9 b^{2} + 6 a b$