Algebra Examples

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

Step 1

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

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

Step 2

Simplify each term.

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Step 2.1

Multiply $a$ by $a^{3}$ by adding the exponents.

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Step 2.1.1

Multiply $a$ by $a^{3}$ .

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.2

Add $1$ and $3$ .

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

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

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Step 2.3.1

Move $a$ .

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

Step 2.3.2

Multiply $a$ by $a$ .

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

Step 2.4

Rewrite using the commutative property of multiplication.

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

Step 2.5

Rewrite using the commutative property of multiplication.

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

Step 2.6

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

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Step 2.6.1

Move $b$ .

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

Step 2.6.2

Multiply $b$ by $b$ .

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

Step 2.7

Rewrite using the commutative property of multiplication.

$a^{4} - 3 a^{2} b - a b^{2} + b a^{3} - 3 b^{2} a - b \cdot b^{2}$

Step 2.8

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

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Step 2.8.1

Move $b^{2}$ .

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

Step 2.8.2

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

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Step 2.8.2.1

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

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

Step 2.8.2.2

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

$a^{4} - 3 a^{2} b - a b^{2} + b a^{3} - 3 b^{2} a - b^{2 + 1}$

Step 2.8.3

Add $2$ and $1$ .

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

Step 3

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

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Step 3.1

Move $b^{2}$ .

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

Step 3.2

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

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

Step 4

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

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

Step 5

Move $- 3 a^{2} b$ .

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

Step 6

Move $- 4 a b^{2}$ .

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

Step 7

Reorder $a^{4}$ and $a^{3} b$ .

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