Algebra Examples

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

Step 1

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 $b = b - c$ .

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

Step 2

Simplify.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply $- - c$ .

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Step 2.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2.2

Multiply $c$ by $1$ .

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

Step 2.3

Subtract $b$ from $b$ .

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

Step 2.4

Add $a$ and $0$ .

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

Step 2.5

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

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

Step 2.6

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

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Apply the distributive property.

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

Step 2.6.3

Apply the distributive property.

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

Step 2.7

Simplify and combine like terms.

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Step 2.7.1

Simplify each term.

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Step 2.7.1.1

Multiply $a$ by $a$ .

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

Step 2.7.1.2

Multiply $b$ by $b$ .

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

Step 2.7.2

Add $a b$ and $b a$ .

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Step 2.7.2.1

Reorder $b$ and $a$ .

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

Step 2.7.2.2

Add $a b$ and $a b$ .

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

Step 2.8

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

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Step 2.8.1

Apply the distributive property.

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

Step 2.8.2

Apply the distributive property.

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

Step 2.8.3

Apply the distributive property.

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

Step 2.9

Simplify each term.

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Step 2.9.1

Rewrite using the commutative property of multiplication.

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

Step 2.9.2

Multiply $b$ by $b$ .

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

Step 2.9.3

Rewrite using the commutative property of multiplication.

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

Step 2.10

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

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

Step 2.11

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

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Step 2.11.1

Apply the distributive property.

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

Step 2.11.2

Apply the distributive property.

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

Step 2.11.3

Apply the distributive property.

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

Step 2.12

Simplify and combine like terms.

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Step 2.12.1

Simplify each term.

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Step 2.12.1.1

Multiply $b$ by $b$ .

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

Step 2.12.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.12.1.3

Rewrite using the commutative property of multiplication.

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

Step 2.12.1.4

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

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Step 2.12.1.4.1

Move $c$ .

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

Step 2.12.1.4.2

Multiply $c$ by $c$ .

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

Step 2.12.1.5

Multiply $- 1$ by $- 1$ .

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

Step 2.12.1.6

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

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

Step 2.12.2

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

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Step 2.12.2.1

Move $c$ .

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

Step 2.12.2.2

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

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

Step 2.13

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

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

Step 2.14

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

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

Step 2.15

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

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

Step 2.16

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

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