Pre-Algebra Examples

$(r^{4} - 16) \div (r - 2)$

Step 1

Rewrite the division as a fraction.

$\frac{r^{4} - 16}{r - 2}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{{(r^{2})}^{2} - 16}{r - 2}$

Step 2.2

Rewrite $16$ as $4^{2}$ .

$\frac{{(r^{2})}^{2} - 4^{2}}{r - 2}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = r^{2}$ and $b = 4$ .

$\frac{(r^{2} + 4) (r^{2} - 4)}{r - 2}$

Step 2.4

Simplify.

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Step 2.4.1

Rewrite $4$ as $2^{2}$ .

$\frac{(r^{2} + 4) (r^{2} - 2^{2})}{r - 2}$

Step 2.4.2

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

$\frac{(r^{2} + 4) (r + 2) (r - 2)}{r - 2}$

Step 3

Cancel the common factor of $r - 2$ .

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Step 3.1

Cancel the common factor.

$\frac{(r^{2} + 4) (r + 2) (r - 2)}{r - 2}$

Step 3.2

Divide $(r^{2} + 4) (r + 2)$ by $1$ .

$(r^{2} + 4) (r + 2)$

Step 4

Expand $(r^{2} + 4) (r + 2)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$r^{2} (r + 2) + 4 (r + 2)$

Step 4.2

Apply the distributive property.

$r^{2} r + r^{2} \cdot 2 + 4 (r + 2)$

Step 4.3

Apply the distributive property.

$r^{2} r + r^{2} \cdot 2 + 4 r + 4 \cdot 2$

Step 5

Simplify each term.

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Step 5.1

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

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Step 5.1.1

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

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Step 5.1.1.1

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

$r^{2} r^{1} + r^{2} \cdot 2 + 4 r + 4 \cdot 2$

Step 5.1.1.2

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

$r^{2 + 1} + r^{2} \cdot 2 + 4 r + 4 \cdot 2$

Step 5.1.2

Add $2$ and $1$ .

$r^{3} + r^{2} \cdot 2 + 4 r + 4 \cdot 2$

Step 5.2

Move $2$ to the left of $r^{2}$ .

$r^{3} + 2 \cdot r^{2} + 4 r + 4 \cdot 2$

Step 5.3

Multiply $4$ by $2$ .

$r^{3} + 2 r^{2} + 4 r + 8$