Algebra Examples

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

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 $b = b$ .

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

Step 2

Simplify.

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Step 2.1

Subtract $b$ from $b$ .

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

Step 2.2

Add $a$ and $0$ .

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

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Apply the distributive property.

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

Step 2.4.3

Apply the distributive property.

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

Step 2.5

Simplify and combine like terms.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

Multiply $a$ by $a$ .

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

Step 2.5.1.2

Multiply $b$ by $b$ .

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

Step 2.5.2

Add $a b$ and $b a$ .

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Step 2.5.2.1

Reorder $b$ and $a$ .

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

Step 2.5.2.2

Add $a b$ and $a b$ .

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

Step 2.6

Apply the distributive property.

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

Step 2.7

Multiply $b$ by $b$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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