Finite Math Examples

$(a + b) (2 a + b) (a - 2 b)$

Step 1

Expand $(a + b) (2 a + b)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$(a (2 a + b) + b (2 a + b)) (a - 2 b)$

Step 1.2

Apply the distributive property.

$(a (2 a) + a b + b (2 a + b)) (a - 2 b)$

Step 1.3

Apply the distributive property.

$(a (2 a) + a b + b (2 a) + b \cdot b) (a - 2 b)$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

$(2 a \cdot a + a b + b (2 a) + b \cdot b) (a - 2 b)$

Step 2.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 2.1.2.1

Move $a$ .

$(2 (a \cdot a) + a b + b (2 a) + b \cdot b) (a - 2 b)$

Step 2.1.2.2

Multiply $a$ by $a$ .

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

Step 2.1.3

Rewrite using the commutative property of multiplication.

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

Step 2.1.4

Multiply $b$ by $b$ .

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

Step 2.2

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

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Step 2.2.1

Move $b$ .

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

Step 2.2.2

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

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

Step 3

Expand $(2 a^{2} + 3 a b + b^{2}) (a - 2 b)$ by multiplying each term in the first expression by each term in the second expression.

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

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

Move $a$ .

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

Step 4.1.1.2

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

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Step 4.1.1.2.1

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

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

Step 4.1.1.2.2

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

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

Step 4.1.1.3

Add $1$ and $2$ .

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

Step 4.1.2

Rewrite using the commutative property of multiplication.

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

Step 4.1.3

Multiply $2$ by $- 2$ .

$2 a^{3} - 4 a^{2} b + 3 a b a + 3 a b (- 2 b) + b^{2} a + b^{2} (- 2 b)$

Step 4.1.4

Multiply $a$ by $a$ by adding the exponents.

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Step 4.1.4.1

Move $a$ .

$2 a^{3} - 4 a^{2} b + 3 (a \cdot a) b + 3 a b (- 2 b) + b^{2} a + b^{2} (- 2 b)$

Step 4.1.4.2

Multiply $a$ by $a$ .

$2 a^{3} - 4 a^{2} b + 3 a^{2} b + 3 a b (- 2 b) + b^{2} a + b^{2} (- 2 b)$

Step 4.1.5

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

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Step 4.1.5.1

Move $b$ .

$2 a^{3} - 4 a^{2} b + 3 a^{2} b + 3 a (b \cdot b) \cdot - 2 + b^{2} a + b^{2} (- 2 b)$

Step 4.1.5.2

Multiply $b$ by $b$ .

$2 a^{3} - 4 a^{2} b + 3 a^{2} b + 3 a b^{2} \cdot - 2 + b^{2} a + b^{2} (- 2 b)$

Step 4.1.6

Multiply $- 2$ by $3$ .

$2 a^{3} - 4 a^{2} b + 3 a^{2} b - 6 a b^{2} + b^{2} a + b^{2} (- 2 b)$

Step 4.1.7

Rewrite using the commutative property of multiplication.

$2 a^{3} - 4 a^{2} b + 3 a^{2} b - 6 a b^{2} + b^{2} a - 2 b^{2} b$

Step 4.1.8

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

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Step 4.1.8.1

Move $b$ .

$2 a^{3} - 4 a^{2} b + 3 a^{2} b - 6 a b^{2} + b^{2} a - 2 (b \cdot b^{2})$

Step 4.1.8.2

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

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Step 4.1.8.2.1

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

$2 a^{3} - 4 a^{2} b + 3 a^{2} b - 6 a b^{2} + b^{2} a - 2 (b^{1} b^{2})$

Step 4.1.8.2.2

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

$2 a^{3} - 4 a^{2} b + 3 a^{2} b - 6 a b^{2} + b^{2} a - 2 b^{1 + 2}$

Step 4.1.8.3

Add $1$ and $2$ .

$2 a^{3} - 4 a^{2} b + 3 a^{2} b - 6 a b^{2} + b^{2} a - 2 b^{3}$

Step 4.2

Add $- 4 a^{2} b$ and $3 a^{2} b$ .

$2 a^{3} - a^{2} b - 6 a b^{2} + b^{2} a - 2 b^{3}$

Step 5

Add $- 6 a b^{2}$ and $b^{2} a$ .

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Step 5.1

Reorder $b^{2}$ and $a$ .

$2 a^{3} - a^{2} b - 6 a b^{2} + a b^{2} - 2 b^{3}$

Step 5.2

Add $- 6 a b^{2}$ and $a b^{2}$ .

$2 a^{3} - a^{2} b - 5 a b^{2} - 2 b^{3}$