Algebra Examples

$(a + b) (3 a - b) (2 a + 7 b)$

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$a x^{2} + b x + c$

Step 2

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

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Step 2.1

Apply the distributive property.

$(a (3 a - b) + b (3 a - b)) (2 a + 7 b)$

Step 2.2

Apply the distributive property.

$(a (3 a) + a (- b) + b (3 a - b)) (2 a + 7 b)$

Step 2.3

Apply the distributive property.

$(a (3 a) + a (- b) + b (3 a) + b (- b)) (2 a + 7 b)$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$(3 a \cdot a + a (- b) + b (3 a) + b (- b)) (2 a + 7 b)$

Step 3.1.2

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

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Step 3.1.2.1

Move $a$ .

$(3 (a \cdot a) + a (- b) + b (3 a) + b (- b)) (2 a + 7 b)$

Step 3.1.2.2

Multiply $a$ by $a$ .

$(3 a^{2} + a (- b) + b (3 a) + b (- b)) (2 a + 7 b)$

Step 3.1.3

Rewrite using the commutative property of multiplication.

$(3 a^{2} - a b + b (3 a) + b (- b)) (2 a + 7 b)$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$(3 a^{2} - a b + 3 b a + b (- b)) (2 a + 7 b)$

Step 3.1.5

Rewrite using the commutative property of multiplication.

$(3 a^{2} - a b + 3 b a - b \cdot b) (2 a + 7 b)$

Step 3.1.6

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

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Step 3.1.6.1

Move $b$ .

$(3 a^{2} - a b + 3 b a - (b \cdot b)) (2 a + 7 b)$

Step 3.1.6.2

Multiply $b$ by $b$ .

$(3 a^{2} - a b + 3 b a - b^{2}) (2 a + 7 b)$

Step 3.2

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

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Step 3.2.1

Move $b$ .

$(3 a^{2} - a b + 3 a b - b^{2}) (2 a + 7 b)$

Step 3.2.2

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

$(3 a^{2} + 2 a b - b^{2}) (2 a + 7 b)$

Step 4

Expand $(3 a^{2} + 2 a b - b^{2}) (2 a + 7 b)$ by multiplying each term in the first expression by each term in the second expression.

$3 a^{2} (2 a) + 3 a^{2} (7 b) + 2 a b (2 a) + 2 a b (7 b) - b^{2} (2 a) - b^{2} (7 b)$

Step 5

Simplify terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Rewrite using the commutative property of multiplication.

$3 \cdot 2 a^{2} a + 3 a^{2} (7 b) + 2 a b (2 a) + 2 a b (7 b) - b^{2} (2 a) - b^{2} (7 b)$

Step 5.1.2

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

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Step 5.1.2.1

Move $a$ .

$3 \cdot 2 (a \cdot a^{2}) + 3 a^{2} (7 b) + 2 a b (2 a) + 2 a b (7 b) - b^{2} (2 a) - b^{2} (7 b)$

Step 5.1.2.2

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

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Step 5.1.2.2.1

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

$3 \cdot 2 (a^{1} a^{2}) + 3 a^{2} (7 b) + 2 a b (2 a) + 2 a b (7 b) - b^{2} (2 a) - b^{2} (7 b)$

Step 5.1.2.2.2

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

$3 \cdot 2 a^{1 + 2} + 3 a^{2} (7 b) + 2 a b (2 a) + 2 a b (7 b) - b^{2} (2 a) - b^{2} (7 b)$

Step 5.1.2.3

Add $1$ and $2$ .

$3 \cdot 2 a^{3} + 3 a^{2} (7 b) + 2 a b (2 a) + 2 a b (7 b) - b^{2} (2 a) - b^{2} (7 b)$

Step 5.1.3

Multiply $3$ by $2$ .

$6 a^{3} + 3 a^{2} (7 b) + 2 a b (2 a) + 2 a b (7 b) - b^{2} (2 a) - b^{2} (7 b)$

Step 5.1.4

Rewrite using the commutative property of multiplication.

$6 a^{3} + 3 \cdot 7 a^{2} b + 2 a b (2 a) + 2 a b (7 b) - b^{2} (2 a) - b^{2} (7 b)$

Step 5.1.5

Multiply $3$ by $7$ .

$6 a^{3} + 21 a^{2} b + 2 a b (2 a) + 2 a b (7 b) - b^{2} (2 a) - b^{2} (7 b)$

Step 5.1.6

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

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Step 5.1.6.1

Move $a$ .

$6 a^{3} + 21 a^{2} b + 2 (a \cdot a) b \cdot 2 + 2 a b (7 b) - b^{2} (2 a) - b^{2} (7 b)$

Step 5.1.6.2

Multiply $a$ by $a$ .

$6 a^{3} + 21 a^{2} b + 2 a^{2} b \cdot 2 + 2 a b (7 b) - b^{2} (2 a) - b^{2} (7 b)$

Step 5.1.7

Multiply $2$ by $2$ .

$6 a^{3} + 21 a^{2} b + 4 a^{2} b + 2 a b (7 b) - b^{2} (2 a) - b^{2} (7 b)$

Step 5.1.8

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

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Step 5.1.8.1

Move $b$ .

$6 a^{3} + 21 a^{2} b + 4 a^{2} b + 2 a (b \cdot b) \cdot 7 - b^{2} (2 a) - b^{2} (7 b)$

Step 5.1.8.2

Multiply $b$ by $b$ .

$6 a^{3} + 21 a^{2} b + 4 a^{2} b + 2 a b^{2} \cdot 7 - b^{2} (2 a) - b^{2} (7 b)$

Step 5.1.9

Multiply $7$ by $2$ .

$6 a^{3} + 21 a^{2} b + 4 a^{2} b + 14 a b^{2} - b^{2} (2 a) - b^{2} (7 b)$

Step 5.1.10

Rewrite using the commutative property of multiplication.

$6 a^{3} + 21 a^{2} b + 4 a^{2} b + 14 a b^{2} - 1 \cdot 2 b^{2} a - b^{2} (7 b)$

Step 5.1.11

Multiply $- 1$ by $2$ .

$6 a^{3} + 21 a^{2} b + 4 a^{2} b + 14 a b^{2} - 2 b^{2} a - b^{2} (7 b)$

Step 5.1.12

Rewrite using the commutative property of multiplication.

$6 a^{3} + 21 a^{2} b + 4 a^{2} b + 14 a b^{2} - 2 b^{2} a - 1 \cdot 7 b^{2} b$

Step 5.1.13

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

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Step 5.1.13.1

Move $b$ .

$6 a^{3} + 21 a^{2} b + 4 a^{2} b + 14 a b^{2} - 2 b^{2} a - 1 \cdot 7 (b \cdot b^{2})$

Step 5.1.13.2

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

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Step 5.1.13.2.1

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

$6 a^{3} + 21 a^{2} b + 4 a^{2} b + 14 a b^{2} - 2 b^{2} a - 1 \cdot 7 (b^{1} b^{2})$

Step 5.1.13.2.2

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

$6 a^{3} + 21 a^{2} b + 4 a^{2} b + 14 a b^{2} - 2 b^{2} a - 1 \cdot 7 b^{1 + 2}$

Step 5.1.13.3

Add $1$ and $2$ .

$6 a^{3} + 21 a^{2} b + 4 a^{2} b + 14 a b^{2} - 2 b^{2} a - 1 \cdot 7 b^{3}$

Step 5.1.14

Multiply $- 1$ by $7$ .

$6 a^{3} + 21 a^{2} b + 4 a^{2} b + 14 a b^{2} - 2 b^{2} a - 7 b^{3}$

Step 5.2

Add $21 a^{2} b$ and $4 a^{2} b$ .

$6 a^{3} + 25 a^{2} b + 14 a b^{2} - 2 b^{2} a - 7 b^{3}$

Step 6

Subtract $2 b^{2} a$ from $14 a b^{2}$ .

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Step 6.1

Move $b^{2}$ .

$6 a^{3} + 25 a^{2} b + 14 a b^{2} - 2 a b^{2} - 7 b^{3}$

Step 6.2

Subtract $2 a b^{2}$ from $14 a b^{2}$ .

$6 a^{3} + 25 a^{2} b + 12 a b^{2} - 7 b^{3}$