Pre-Algebra Examples

$\frac{Q^{4} - 1}{Q + 1}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $Q^{4}$ as ${(Q^{2})}^{2}$ .

$\frac{{(Q^{2})}^{2} - 1}{Q + 1}$

Step 1.2

Rewrite $1$ as $1^{2}$ .

$\frac{{(Q^{2})}^{2} - 1^{2}}{Q + 1}$

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = Q^{2}$ and $b = 1$ .

$\frac{(Q^{2} + 1) (Q^{2} - 1)}{Q + 1}$

Step 1.4

Simplify.

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Step 1.4.1

Rewrite $1$ as $1^{2}$ .

$\frac{(Q^{2} + 1) (Q^{2} - 1^{2})}{Q + 1}$

Step 1.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = Q$ and $b = 1$ .

$\frac{(Q^{2} + 1) (Q + 1) (Q - 1)}{Q + 1}$

Step 2

Cancel the common factor of $Q + 1$ .

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Step 2.1

Cancel the common factor.

$\frac{(Q^{2} + 1) (Q + 1) (Q - 1)}{Q + 1}$

Step 2.2

Divide $(Q^{2} + 1) (Q - 1)$ by $1$ .

$(Q^{2} + 1) (Q - 1)$

Step 3

Expand $(Q^{2} + 1) (Q - 1)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$Q^{2} (Q - 1) + 1 (Q - 1)$

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

$Q^{2} Q + Q^{2} \cdot - 1 + 1 Q + 1 \cdot - 1$

Step 4

Simplify each term.

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Step 4.1

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

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Step 4.1.1

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

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Step 4.1.1.1

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

$Q^{2} Q^{1} + Q^{2} \cdot - 1 + 1 Q + 1 \cdot - 1$

Step 4.1.1.2

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

$Q^{2 + 1} + Q^{2} \cdot - 1 + 1 Q + 1 \cdot - 1$

Step 4.1.2

Add $2$ and $1$ .

$Q^{3} + Q^{2} \cdot - 1 + 1 Q + 1 \cdot - 1$

Step 4.2

Move $- 1$ to the left of $Q^{2}$ .

$Q^{3} - 1 \cdot Q^{2} + 1 Q + 1 \cdot - 1$

Step 4.3

Rewrite $- 1 Q^{2}$ as $- Q^{2}$ .

$Q^{3} - Q^{2} + 1 Q + 1 \cdot - 1$

Step 4.4

Multiply $Q$ by $1$ .

$Q^{3} - Q^{2} + Q + 1 \cdot - 1$

Step 4.5

Multiply $- 1$ by $1$ .

$Q^{3} - Q^{2} + Q - 1$