Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Simplify each term.

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Step 1.2.1

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

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Step 1.2.1.1

Move $a$ .

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

Step 1.2.1.2

Multiply $a$ by $a$ .

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

Step 1.2.2

Rewrite using the commutative property of multiplication.

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

Step 1.2.3

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

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Step 1.2.3.1

Move $b^{2}$ .

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

Step 1.2.3.2

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

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Step 1.2.3.2.1

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

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

Step 1.2.3.2.2

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

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

Step 1.2.3.3

Add $2$ and $1$ .

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

Step 1.2.4

Move $2$ to the left of $a b$ .

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

Step 1.2.5

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

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Step 1.2.5.1

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

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Step 1.2.5.1.1

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

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

Step 1.2.5.1.2

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

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

Step 1.2.5.2

Add $2$ and $1$ .

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

Step 1.2.6

Rewrite using the commutative property of multiplication.

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

Step 1.2.7

Move $2$ to the left of $a^{2}$ .

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

Step 1.2.8

Rewrite using the commutative property of multiplication.

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

Step 1.2.9

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

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Step 1.2.9.1

Move $b^{2}$ .

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

Step 1.2.9.2

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

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Step 1.2.9.2.1

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

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

Step 1.2.9.2.2

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

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

Step 1.2.9.3

Add $2$ and $1$ .

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

Step 1.2.10

Multiply $- 1$ by $- 1$ .

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

Step 1.2.11

Multiply $b^{3}$ by $1$ .

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

Step 1.2.12

Multiply $2$ by $- 1$ .

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

Step 1.3

Subtract $b a$ from $2 a b$ .

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Step 1.3.1

Move $b$ .

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

Step 1.3.2

Subtract $a b$ from $2 a b$ .

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

Step 1.4

Multiply $a$ by $1$ .

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

Step 1.5

Apply the distributive property.

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

Step 1.6

Multiply $- 1$ by $- 2$ .

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

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $a^{2} b - a b^{3} + a^{3} - a^{2} b^{2} + 2 a^{2} + a b + b^{3} - 2 b - 2 a^{2} + 2 b$ .

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Step 2.1.1

Subtract $2 a^{2}$ from $2 a^{2}$ .

$a^{2} b - a b^{3} + a^{3} - a^{2} b^{2} + a b + b^{3} - 2 b + 0 + 2 b$

Step 2.1.2

Add $a^{2} b - a b^{3} + a^{3} - a^{2} b^{2} + a b + b^{3} - 2 b$ and $0$ .

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

Step 2.1.3

Add $- 2 b$ and $2 b$ .

$a^{2} b - a b^{3} + a^{3} - a^{2} b^{2} + a b + b^{3} + 0$

Step 2.1.4

Add $a^{2} b - a b^{3} + a^{3} - a^{2} b^{2} + a b + b^{3}$ and $0$ .

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

Step 2.2

Simplify the expression.

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Step 2.2.1

Move $a b$ .

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

Step 2.2.2

Move $a^{2} b$ .

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

Step 2.2.3

Move $a^{3}$ .

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

Step 2.2.4

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

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