Algebra Examples

$(p + q) (4 p^{2} - p - 8 q^{2} - q)$

Step 1

Expand $(p + q) (4 p^{2} - p - 8 q^{2} - q)$ by multiplying each term in the first expression by each term in the second expression.

$p (4 p^{2}) + p (- p) + p (- 8 q^{2}) + p (- q) + q (4 p^{2}) + q (- p) + q (- 8 q^{2}) + q (- q)$

Step 2

Simplify each term.

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Step 2.1

Rewrite using the commutative property of multiplication.

$4 p \cdot p^{2} + p (- p) + p (- 8 q^{2}) + p (- q) + q (4 p^{2}) + q (- p) + q (- 8 q^{2}) + q (- q)$

Step 2.2

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

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Step 2.2.1

Move $p^{2}$ .

$4 (p^{2} p) + p (- p) + p (- 8 q^{2}) + p (- q) + q (4 p^{2}) + q (- p) + q (- 8 q^{2}) + q (- q)$

Step 2.2.2

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

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Step 2.2.2.1

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

$4 (p^{2} p^{1}) + p (- p) + p (- 8 q^{2}) + p (- q) + q (4 p^{2}) + q (- p) + q (- 8 q^{2}) + q (- q)$

Step 2.2.2.2

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

$4 p^{2 + 1} + p (- p) + p (- 8 q^{2}) + p (- q) + q (4 p^{2}) + q (- p) + q (- 8 q^{2}) + q (- q)$

Step 2.2.3

Add $2$ and $1$ .

$4 p^{3} + p (- p) + p (- 8 q^{2}) + p (- q) + q (4 p^{2}) + q (- p) + q (- 8 q^{2}) + q (- q)$

Step 2.3

Rewrite using the commutative property of multiplication.

$4 p^{3} - p \cdot p + p (- 8 q^{2}) + p (- q) + q (4 p^{2}) + q (- p) + q (- 8 q^{2}) + q (- q)$

Step 2.4

Multiply $p$ by $p$ by adding the exponents.

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Step 2.4.1

Move $p$ .

$4 p^{3} - (p \cdot p) + p (- 8 q^{2}) + p (- q) + q (4 p^{2}) + q (- p) + q (- 8 q^{2}) + q (- q)$

Step 2.4.2

Multiply $p$ by $p$ .

$4 p^{3} - p^{2} + p (- 8 q^{2}) + p (- q) + q (4 p^{2}) + q (- p) + q (- 8 q^{2}) + q (- q)$

Step 2.5

Rewrite using the commutative property of multiplication.

$4 p^{3} - p^{2} - 8 p q^{2} + p (- q) + q (4 p^{2}) + q (- p) + q (- 8 q^{2}) + q (- q)$

Step 2.6

Rewrite using the commutative property of multiplication.

$4 p^{3} - p^{2} - 8 p q^{2} - p q + q (4 p^{2}) + q (- p) + q (- 8 q^{2}) + q (- q)$

Step 2.7

Rewrite using the commutative property of multiplication.

$4 p^{3} - p^{2} - 8 p q^{2} - p q + 4 q p^{2} + q (- p) + q (- 8 q^{2}) + q (- q)$

Step 2.8

Rewrite using the commutative property of multiplication.

$4 p^{3} - p^{2} - 8 p q^{2} - p q + 4 q p^{2} - q p + q (- 8 q^{2}) + q (- q)$

Step 2.9

Rewrite using the commutative property of multiplication.

$4 p^{3} - p^{2} - 8 p q^{2} - p q + 4 q p^{2} - q p - 8 q \cdot q^{2} + q (- q)$

Step 2.10

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

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Step 2.10.1

Move $q^{2}$ .

$4 p^{3} - p^{2} - 8 p q^{2} - p q + 4 q p^{2} - q p - 8 (q^{2} q) + q (- q)$

Step 2.10.2

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

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Step 2.10.2.1

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

$4 p^{3} - p^{2} - 8 p q^{2} - p q + 4 q p^{2} - q p - 8 (q^{2} q^{1}) + q (- q)$

Step 2.10.2.2

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

$4 p^{3} - p^{2} - 8 p q^{2} - p q + 4 q p^{2} - q p - 8 q^{2 + 1} + q (- q)$

Step 2.10.3

Add $2$ and $1$ .

$4 p^{3} - p^{2} - 8 p q^{2} - p q + 4 q p^{2} - q p - 8 q^{3} + q (- q)$

Step 2.11

Rewrite using the commutative property of multiplication.

$4 p^{3} - p^{2} - 8 p q^{2} - p q + 4 q p^{2} - q p - 8 q^{3} - q \cdot q$

Step 2.12

Multiply $q$ by $q$ by adding the exponents.

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Step 2.12.1

Move $q$ .

$4 p^{3} - p^{2} - 8 p q^{2} - p q + 4 q p^{2} - q p - 8 q^{3} - (q \cdot q)$

Step 2.12.2

Multiply $q$ by $q$ .

$4 p^{3} - p^{2} - 8 p q^{2} - p q + 4 q p^{2} - q p - 8 q^{3} - q^{2}$

Step 3

Subtract $q p$ from $- p q$ .

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Step 3.1

Move $q$ .

$4 p^{3} - p^{2} - 8 p q^{2} - p q - p q + 4 q p^{2} - 8 q^{3} - q^{2}$

Step 3.2

Subtract $p q$ from $- p q$ .

$4 p^{3} - p^{2} - 8 p q^{2} - 2 p q + 4 q p^{2} - 8 q^{3} - q^{2}$