Algebra Examples

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

Step 1

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

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

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Combine the opposite terms in $a \cdot a + a (- b^{3}) + a (- 3 c) - b^{3} a - b^{3} (- b^{3}) - b^{3} (- 3 c) + 3 c a + 3 c (- b^{3}) + 3 c (- 3 c)$ .

Tap for more steps...

Step 2.1.1

Reorder the factors in the terms $a (- 3 c)$ and $3 c a$ .

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

Step 2.1.2

Add $- 3 a c$ and $3 a c$ .

$a \cdot a + a (- b^{3}) - b^{3} a - b^{3} (- b^{3}) - b^{3} (- 3 c) + 0 + 3 c (- b^{3}) + 3 c (- 3 c)$

Step 2.1.3

Add $a \cdot a + a (- b^{3}) - b^{3} a - b^{3} (- b^{3}) - b^{3} (- 3 c)$ and $0$ .

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

Step 2.2

Simplify each term.

Tap for more steps...

Step 2.2.1

Multiply $a$ by $a$ .

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

Step 2.2.2

Rewrite using the commutative property of multiplication.

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

Step 2.2.3

Rewrite using the commutative property of multiplication.

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

Step 2.2.4

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

Tap for more steps...

Step 2.2.4.1

Move $b^{3}$ .

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

Step 2.2.4.2

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

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

Step 2.2.4.3

Add $3$ and $3$ .

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

Step 2.2.5

Multiply $- 1$ by $- 1$ .

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

Step 2.2.6

Multiply $b^{6}$ by $1$ .

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

Step 2.2.7

Rewrite using the commutative property of multiplication.

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

Step 2.2.8

Multiply $- 1$ by $- 3$ .

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

Step 2.2.9

Rewrite using the commutative property of multiplication.

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

Step 2.2.10

Multiply $3$ by $- 1$ .

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

Step 2.2.11

Rewrite using the commutative property of multiplication.

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

Step 2.2.12

Multiply $c$ by $c$ by adding the exponents.

Tap for more steps...

Step 2.2.12.1

Move $c$ .

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

Step 2.2.12.2

Multiply $c$ by $c$ .

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

Step 2.2.13

Multiply $3$ by $- 3$ .

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

Step 2.3

Combine the opposite terms in $a^{2} - a b^{3} - b^{3} a + b^{6} + 3 b^{3} c - 3 c b^{3} - 9 c^{2}$ .

Tap for more steps...

Step 2.3.1

Reorder the factors in the terms $3 b^{3} c$ and $- 3 c b^{3}$ .

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

Step 2.3.2

Subtract $3 b^{3} c$ from $3 b^{3} c$ .

$a^{2} - a b^{3} - b^{3} a + b^{6} + 0 - 9 c^{2}$

Step 2.3.3

Add $a^{2} - a b^{3} - b^{3} a + b^{6}$ and $0$ .

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

Step 3

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

Tap for more steps...

Step 3.1

Move $b^{3}$ .

$a^{2} - a b^{3} - 1 a b^{3} + b^{6} - 9 c^{2}$

Step 3.2

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

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

Step 4

Move $a^{2}$ .

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

Step 5

Reorder $- 2 a b^{3}$ and $b^{6}$ .

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