Pre-Algebra Examples

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

Step 1

Expand $(2 p - 1) (4 p^{2} + 2 p + 1)$ by multiplying each term in the first expression by each term in the second expression.

$2 p (4 p^{2}) + 2 p (2 p) + 2 p \cdot 1 - 1 (4 p^{2}) - 1 (2 p) - 1 \cdot 1$

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $2 p (4 p^{2}) + 2 p (2 p) + 2 p \cdot 1 - 1 (4 p^{2}) - 1 (2 p) - 1 \cdot 1$ .

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Step 2.1.1

Reorder the factors in the terms $2 p \cdot 1$ and $- 1 (2 p)$ .

$2 p (4 p^{2}) + 2 p (2 p) + 1 \cdot 2 p - 1 (4 p^{2}) - 1 \cdot 2 p - 1 \cdot 1$

Step 2.1.2

Subtract $2 p$ from $1 \cdot 2 p$ .

$2 p (4 p^{2}) + 2 p (2 p) - 1 (4 p^{2}) + 0 - 1 \cdot 1$

Step 2.1.3

Add $2 p (4 p^{2}) + 2 p (2 p) - 1 (4 p^{2})$ and $0$ .

$2 p (4 p^{2}) + 2 p (2 p) - 1 (4 p^{2}) - 1 \cdot 1$

Step 2.2

Simplify each term.

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Step 2.2.1

Rewrite using the commutative property of multiplication.

$2 \cdot 4 p \cdot p^{2} + 2 p (2 p) - 1 (4 p^{2}) - 1 \cdot 1$

Step 2.2.2

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

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Step 2.2.2.1

Move $p^{2}$ .

$2 \cdot 4 (p^{2} p) + 2 p (2 p) - 1 (4 p^{2}) - 1 \cdot 1$

Step 2.2.2.2

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

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Step 2.2.2.2.1

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

$2 \cdot 4 (p^{2} p^{1}) + 2 p (2 p) - 1 (4 p^{2}) - 1 \cdot 1$

Step 2.2.2.2.2

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

$2 \cdot 4 p^{2 + 1} + 2 p (2 p) - 1 (4 p^{2}) - 1 \cdot 1$

Step 2.2.2.3

Add $2$ and $1$ .

$2 \cdot 4 p^{3} + 2 p (2 p) - 1 (4 p^{2}) - 1 \cdot 1$

Step 2.2.3

Multiply $2$ by $4$ .

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

Step 2.2.4

Rewrite using the commutative property of multiplication.

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

Step 2.2.5

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

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Step 2.2.5.1

Move $p$ .

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

Step 2.2.5.2

Multiply $p$ by $p$ .

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

Step 2.2.6

Multiply $2$ by $2$ .

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

Step 2.2.7

Multiply $4$ by $- 1$ .

$8 p^{3} + 4 p^{2} - 4 p^{2} - 1 \cdot 1$

Step 2.2.8

Multiply $- 1$ by $1$ .

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

Step 2.3

Combine the opposite terms in $8 p^{3} + 4 p^{2} - 4 p^{2} - 1$ .

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Step 2.3.1

Subtract $4 p^{2}$ from $4 p^{2}$ .

$8 p^{3} + 0 - 1$

Step 2.3.2

Add $8 p^{3}$ and $0$ .

$8 p^{3} - 1$