Algebra Examples

$(f + 2 g) (2 f^{2} + 3 (f g) - g^{2})$

Step 1

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

$f (2 f^{2}) + f (3 f g) + f (- g^{2}) + 2 g (2 f^{2}) + 2 g (3 f g) + 2 g (- g^{2})$

Step 2

Simplify each term.

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Step 2.1

Rewrite using the commutative property of multiplication.

$2 f \cdot f^{2} + f (3 f g) + f (- g^{2}) + 2 g (2 f^{2}) + 2 g (3 f g) + 2 g (- g^{2})$

Step 2.2

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

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Step 2.2.1

Move $f^{2}$ .

$2 (f^{2} f) + f (3 f g) + f (- g^{2}) + 2 g (2 f^{2}) + 2 g (3 f g) + 2 g (- g^{2})$

Step 2.2.2

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

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Step 2.2.2.1

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

$2 (f^{2} f^{1}) + f (3 f g) + f (- g^{2}) + 2 g (2 f^{2}) + 2 g (3 f g) + 2 g (- g^{2})$

Step 2.2.2.2

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

$2 f^{2 + 1} + f (3 f g) + f (- g^{2}) + 2 g (2 f^{2}) + 2 g (3 f g) + 2 g (- g^{2})$

Step 2.2.3

Add $2$ and $1$ .

$2 f^{3} + f (3 f g) + f (- g^{2}) + 2 g (2 f^{2}) + 2 g (3 f g) + 2 g (- g^{2})$

Step 2.3

Rewrite using the commutative property of multiplication.

$2 f^{3} + 3 f (f g) + f (- g^{2}) + 2 g (2 f^{2}) + 2 g (3 f g) + 2 g (- g^{2})$

Step 2.4

Multiply $f$ by $f$ by adding the exponents.

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Step 2.4.1

Move $f$ .

$2 f^{3} + 3 (f \cdot f) g + f (- g^{2}) + 2 g (2 f^{2}) + 2 g (3 f g) + 2 g (- g^{2})$

Step 2.4.2

Multiply $f$ by $f$ .

$2 f^{3} + 3 f^{2} g + f (- g^{2}) + 2 g (2 f^{2}) + 2 g (3 f g) + 2 g (- g^{2})$

Step 2.5

Rewrite using the commutative property of multiplication.

$2 f^{3} + 3 f^{2} g - f g^{2} + 2 g (2 f^{2}) + 2 g (3 f g) + 2 g (- g^{2})$

Step 2.6

Rewrite using the commutative property of multiplication.

$2 f^{3} + 3 f^{2} g - f g^{2} + 2 \cdot 2 g f^{2} + 2 g (3 f g) + 2 g (- g^{2})$

Step 2.7

Multiply $2$ by $2$ .

$2 f^{3} + 3 f^{2} g - f g^{2} + 4 g f^{2} + 2 g (3 f g) + 2 g (- g^{2})$

Step 2.8

Multiply $g$ by $g$ by adding the exponents.

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Step 2.8.1

Move $g$ .

$2 f^{3} + 3 f^{2} g - f g^{2} + 4 g f^{2} + 2 (g \cdot g) (3 f) + 2 g (- g^{2})$

Step 2.8.2

Multiply $g$ by $g$ .

$2 f^{3} + 3 f^{2} g - f g^{2} + 4 g f^{2} + 2 g^{2} (3 f) + 2 g (- g^{2})$

Step 2.9

Rewrite using the commutative property of multiplication.

$2 f^{3} + 3 f^{2} g - f g^{2} + 4 g f^{2} + 2 \cdot 3 g^{2} f + 2 g (- g^{2})$

Step 2.10

Multiply $2$ by $3$ .

$2 f^{3} + 3 f^{2} g - f g^{2} + 4 g f^{2} + 6 g^{2} f + 2 g (- g^{2})$

Step 2.11

Rewrite using the commutative property of multiplication.

$2 f^{3} + 3 f^{2} g - f g^{2} + 4 g f^{2} + 6 g^{2} f + 2 \cdot - 1 g \cdot g^{2}$

Step 2.12

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

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Step 2.12.1

Move $g^{2}$ .

$2 f^{3} + 3 f^{2} g - f g^{2} + 4 g f^{2} + 6 g^{2} f + 2 \cdot - 1 (g^{2} g)$

Step 2.12.2

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

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Step 2.12.2.1

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

$2 f^{3} + 3 f^{2} g - f g^{2} + 4 g f^{2} + 6 g^{2} f + 2 \cdot - 1 (g^{2} g^{1})$

Step 2.12.2.2

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

$2 f^{3} + 3 f^{2} g - f g^{2} + 4 g f^{2} + 6 g^{2} f + 2 \cdot - 1 g^{2 + 1}$

Step 2.12.3

Add $2$ and $1$ .

$2 f^{3} + 3 f^{2} g - f g^{2} + 4 g f^{2} + 6 g^{2} f + 2 \cdot - 1 g^{3}$

Step 2.13

Multiply $2$ by $- 1$ .

$2 f^{3} + 3 f^{2} g - f g^{2} + 4 g f^{2} + 6 g^{2} f - 2 g^{3}$

Step 3

Add $3 f^{2} g$ and $4 g f^{2}$ .

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Step 3.1

Move $g$ .

$2 f^{3} + 3 f^{2} g + 4 f^{2} g - f g^{2} + 6 g^{2} f - 2 g^{3}$

Step 3.2

Add $3 f^{2} g$ and $4 f^{2} g$ .

$2 f^{3} + 7 f^{2} g - f g^{2} + 6 g^{2} f - 2 g^{3}$

Step 4

Add $- f g^{2}$ and $6 g^{2} f$ .

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Step 4.1

Move $g^{2}$ .

$2 f^{3} + 7 f^{2} g - f g^{2} + 6 f g^{2} - 2 g^{3}$

Step 4.2

Add $- f g^{2}$ and $6 f g^{2}$ .

$2 f^{3} + 7 f^{2} g + 5 f g^{2} - 2 g^{3}$