Factor (a+b)^3-(b-c)^3

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

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = a + b$ and $b = b - c$ .

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

Simplify.

Apply the distributive property.

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

Multiply $- - c$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $c$ by $1$ .

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

Subtract $b$ from $b$ .

$(a + 0 + c) ({(a + b)}^{2} + (a + b) (b - c) + {(b - c)}^{2})$

Add $a$ and $0$ .

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

Rewrite ${(a + b)}^{2}$ as $(a + b) (a + b)$ .

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

Expand $(a + b) (a + b)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $a$ by $a$ .

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

Multiply $b$ by $b$ .

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

Add $a b$ and $b a$ .

Reorder $b$ and $a$ .

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

Add $a b$ and $a b$ .

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

Expand $(a + b) (b - c)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify each term.

Rewrite using the commutative property of multiplication.

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

Multiply $b$ by $b$ .

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

Rewrite using the commutative property of multiplication.

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

Rewrite ${(b - c)}^{2}$ as $(b - c) (b - c)$ .

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

Expand $(b - c) (b - c)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $b$ by $b$ .

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

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

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

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

Move $c$ .

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

Multiply $c$ by $c$ .

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

Multiply $- 1$ by $- 1$ .

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

Multiply $c^{2}$ by $1$ .

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

Subtract $c b$ from $- b c$ .

Move $c$ .

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

Subtract $b c$ from $- b c$ .

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

Add $2 a b$ and $a b$ .

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

Add $b^{2}$ and $b^{2}$ .

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

Add $2 b^{2}$ and $b^{2}$ .

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

Subtract $2 b c$ from $- b c$ .

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

Factor (a+b)^3-(b-c)^3

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