Algebra Examples

$(3 a + b) (3 a - b) (2 a + 5 b)$

Step 1

Expand $(3 a + b) (3 a - b)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$(3 a (3 a - b) + b (3 a - b)) (2 a + 5 b)$

Step 1.2

Apply the distributive property.

$(3 a (3 a) + 3 a (- b) + b (3 a - b)) (2 a + 5 b)$

Step 1.3

Apply the distributive property.

$(3 a (3 a) + 3 a (- b) + b (3 a) + b (- b)) (2 a + 5 b)$

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.

$(3 \cdot 3 a \cdot a + 3 a (- b) + b (3 a) + b (- b)) (2 a + 5 b)$

Step 2.1.2

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

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Step 2.1.2.1

Move $a$ .

$(3 \cdot 3 (a \cdot a) + 3 a (- b) + b (3 a) + b (- b)) (2 a + 5 b)$

Step 2.1.2.2

Multiply $a$ by $a$ .

$(3 \cdot 3 a^{2} + 3 a (- b) + b (3 a) + b (- b)) (2 a + 5 b)$

Step 2.1.3

Multiply $3$ by $3$ .

$(9 a^{2} + 3 a (- b) + b (3 a) + b (- b)) (2 a + 5 b)$

Step 2.1.4

Rewrite using the commutative property of multiplication.

$(9 a^{2} + 3 \cdot - 1 a b + b (3 a) + b (- b)) (2 a + 5 b)$

Step 2.1.5

Multiply $3$ by $- 1$ .

$(9 a^{2} - 3 a b + b (3 a) + b (- b)) (2 a + 5 b)$

Step 2.1.6

Rewrite using the commutative property of multiplication.

$(9 a^{2} - 3 a b + 3 b a + b (- b)) (2 a + 5 b)$

Step 2.1.7

Rewrite using the commutative property of multiplication.

$(9 a^{2} - 3 a b + 3 b a - b \cdot b) (2 a + 5 b)$

Step 2.1.8

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

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Step 2.1.8.1

Move $b$ .

$(9 a^{2} - 3 a b + 3 b a - (b \cdot b)) (2 a + 5 b)$

Step 2.1.8.2

Multiply $b$ by $b$ .

$(9 a^{2} - 3 a b + 3 b a - b^{2}) (2 a + 5 b)$

Step 2.2

Add $- 3 a b$ and $3 b a$ .

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Step 2.2.1

Move $b$ .

$(9 a^{2} - 3 a b + 3 a b - b^{2}) (2 a + 5 b)$

Step 2.2.2

Add $- 3 a b$ and $3 a b$ .

$(9 a^{2} + 0 - b^{2}) (2 a + 5 b)$

Step 2.3

Add $9 a^{2}$ and $0$ .

$(9 a^{2} - b^{2}) (2 a + 5 b)$

Step 3

Expand $(9 a^{2} - b^{2}) (2 a + 5 b)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$9 a^{2} (2 a + 5 b) - b^{2} (2 a + 5 b)$

Step 3.2

Apply the distributive property.

$9 a^{2} (2 a) + 9 a^{2} (5 b) - b^{2} (2 a + 5 b)$

Step 3.3

Apply the distributive property.

$9 a^{2} (2 a) + 9 a^{2} (5 b) - b^{2} (2 a) - b^{2} (5 b)$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite using the commutative property of multiplication.

$9 \cdot 2 a^{2} a + 9 a^{2} (5 b) - b^{2} (2 a) - b^{2} (5 b)$

Step 4.1.2

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

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Step 4.1.2.1

Move $a$ .

$9 \cdot 2 (a \cdot a^{2}) + 9 a^{2} (5 b) - b^{2} (2 a) - b^{2} (5 b)$

Step 4.1.2.2

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

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Step 4.1.2.2.1

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

$9 \cdot 2 (a^{1} a^{2}) + 9 a^{2} (5 b) - b^{2} (2 a) - b^{2} (5 b)$

Step 4.1.2.2.2

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

$9 \cdot 2 a^{1 + 2} + 9 a^{2} (5 b) - b^{2} (2 a) - b^{2} (5 b)$

Step 4.1.2.3

Add $1$ and $2$ .

$9 \cdot 2 a^{3} + 9 a^{2} (5 b) - b^{2} (2 a) - b^{2} (5 b)$

Step 4.1.3

Multiply $9$ by $2$ .

$18 a^{3} + 9 a^{2} (5 b) - b^{2} (2 a) - b^{2} (5 b)$

Step 4.1.4

Rewrite using the commutative property of multiplication.

$18 a^{3} + 9 \cdot 5 a^{2} b - b^{2} (2 a) - b^{2} (5 b)$

Step 4.1.5

Multiply $9$ by $5$ .

$18 a^{3} + 45 a^{2} b - b^{2} (2 a) - b^{2} (5 b)$

Step 4.1.6

Rewrite using the commutative property of multiplication.

$18 a^{3} + 45 a^{2} b - 1 \cdot 2 b^{2} a - b^{2} (5 b)$

Step 4.1.7

Multiply $- 1$ by $2$ .

$18 a^{3} + 45 a^{2} b - 2 b^{2} a - b^{2} (5 b)$

Step 4.1.8

Rewrite using the commutative property of multiplication.

$18 a^{3} + 45 a^{2} b - 2 b^{2} a - 1 \cdot 5 b^{2} b$

Step 4.1.9

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

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Step 4.1.9.1

Move $b$ .

$18 a^{3} + 45 a^{2} b - 2 b^{2} a - 1 \cdot 5 (b \cdot b^{2})$

Step 4.1.9.2

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

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Step 4.1.9.2.1

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

$18 a^{3} + 45 a^{2} b - 2 b^{2} a - 1 \cdot 5 (b^{1} b^{2})$

Step 4.1.9.2.2

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

$18 a^{3} + 45 a^{2} b - 2 b^{2} a - 1 \cdot 5 b^{1 + 2}$

Step 4.1.9.3

Add $1$ and $2$ .

$18 a^{3} + 45 a^{2} b - 2 b^{2} a - 1 \cdot 5 b^{3}$

Step 4.1.10

Multiply $- 1$ by $5$ .

$18 a^{3} + 45 a^{2} b - 2 b^{2} a - 5 b^{3}$

Step 4.2

Move $b^{2}$ .

$18 a^{3} + 45 a^{2} b - 2 a b^{2} - 5 b^{3}$