Pre-Algebra Examples

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

Step 1

Regroup terms.

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

Step 2

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 $a = a$ and $b = b$ .

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

Step 3

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

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

Step 4

Factor $a - b$ out of $(a - b) (a^{2} + a b + b^{2}) + (a + b) (a - b)$ .

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Step 4.1

Factor $a - b$ out of $(a + b) (a - b)$ .

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

Step 4.2

Factor $a - b$ out of $(a - b) (a^{2} + a b + b^{2}) + (a - b) (a + b)$ .

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

Step 5

Factor $a - b$ out of ${(a - b)}^{3} + (a - b) (a^{2} + a b + b^{2} + a + b)$ .

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Step 5.1

Factor $a - b$ out of ${(a - b)}^{3}$ .

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

Step 5.2

Factor $a - b$ out of $(a - b) {(a - b)}^{2} + (a - b) (a^{2} + a b + b^{2} + a + b)$ .

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

Step 6

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

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

Step 7

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

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Apply the distributive property.

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

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $a$ by $a$ .

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

Step 8.1.2

Rewrite using the commutative property of multiplication.

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

Step 8.1.3

Rewrite using the commutative property of multiplication.

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

Step 8.1.4

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

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Step 8.1.4.1

Move $b$ .

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

Step 8.1.4.2

Multiply $b$ by $b$ .

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

Step 8.1.5

Multiply $- 1$ by $- 1$ .

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

Step 8.1.6

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

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

Step 8.2

Subtract $b a$ from $- a b$ .

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Step 8.2.1

Move $b$ .

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

Step 8.2.2

Subtract $a b$ from $- a b$ .

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

Step 9

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

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

Step 10

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

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

Step 11

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

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