Algebra Examples

$(a - b) (a + b) (a - 3 b)$

Step 1

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

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Step 1.1

Apply the distributive property.

$(a (a + b) - b (a + b)) (a - 3 b)$

Step 1.2

Apply the distributive property.

$(a \cdot a + a b - b (a + b)) (a - 3 b)$

Step 1.3

Apply the distributive property.

$(a \cdot a + a b - b a - b \cdot b) (a - 3 b)$

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $a \cdot a + a b - b a - b \cdot b$ .

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Step 2.1.1

Reorder the factors in the terms $a b$ and $- b a$ .

$(a \cdot a + a b - a b - b \cdot b) (a - 3 b)$

Step 2.1.2

Subtract $a b$ from $a b$ .

$(a \cdot a + 0 - b \cdot b) (a - 3 b)$

Step 2.1.3

Add $a \cdot a$ and $0$ .

$(a \cdot a - b \cdot b) (a - 3 b)$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $a$ by $a$ .

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

Step 2.2.2

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

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Step 2.2.2.1

Move $b$ .

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

Step 2.2.2.2

Multiply $b$ by $b$ .

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

Step 3

Expand $(a^{2} - b^{2}) (a - 3 b)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $a^{2}$ by $a$ by adding the exponents.

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Step 4.1.1.1

Multiply $a^{2}$ by $a$ .

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Step 4.1.1.1.1

Raise $a$ to the power of $1$ .

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

Step 4.1.1.1.2

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

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

Step 4.1.1.2

Add $2$ and $1$ .

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

Step 4.1.2

Rewrite using the commutative property of multiplication.

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

Step 4.1.3

Rewrite using the commutative property of multiplication.

$a^{3} - 3 a^{2} b - b^{2} a - 1 \cdot - 3 b^{2} b$

Step 4.1.4

Multiply $b^{2}$ by $b$ by adding the exponents.

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Step 4.1.4.1

Move $b$ .

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

Step 4.1.4.2

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

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Step 4.1.4.2.1

Raise $b$ to the power of $1$ .

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

Step 4.1.4.2.2

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

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

Step 4.1.4.3

Add $1$ and $2$ .

$a^{3} - 3 a^{2} b - b^{2} a - 1 \cdot - 3 b^{3}$

Step 4.1.5

Multiply $- 1$ by $- 3$ .

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

Step 4.2

Move $b^{2}$ .

$a^{3} - 3 a^{2} b - 1 a b^{2} + 3 b^{3}$