Pre-Algebra Examples

$(x^{4} - 81) \div (x - 3)$

Step 1

Rewrite the division as a fraction.

$\frac{x^{4} - 81}{x - 3}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{{(x^{2})}^{2} - 81}{x - 3}$

Step 2.2

Rewrite $81$ as $9^{2}$ .

$\frac{{(x^{2})}^{2} - 9^{2}}{x - 3}$

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 $a = x^{2}$ and $b = 9$ .

$\frac{(x^{2} + 9) (x^{2} - 9)}{x - 3}$

Step 2.4

Simplify.

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Step 2.4.1

Rewrite $9$ as $3^{2}$ .

$\frac{(x^{2} + 9) (x^{2} - 3^{2})}{x - 3}$

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 $a = x$ and $b = 3$ .

$\frac{(x^{2} + 9) (x + 3) (x - 3)}{x - 3}$

Step 3

Cancel the common factor of $x - 3$ .

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Step 3.1

Cancel the common factor.

$\frac{(x^{2} + 9) (x + 3) (x - 3)}{x - 3}$

Step 3.2

Divide $(x^{2} + 9) (x + 3)$ by $1$ .

$(x^{2} + 9) (x + 3)$

Step 4

Expand $(x^{2} + 9) (x + 3)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$x^{2} (x + 3) + 9 (x + 3)$

Step 4.2

Apply the distributive property.

$x^{2} x + x^{2} \cdot 3 + 9 (x + 3)$

Step 4.3

Apply the distributive property.

$x^{2} x + x^{2} \cdot 3 + 9 x + 9 \cdot 3$

Step 5

Simplify each term.

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Step 5.1

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

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Step 5.1.1

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

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Step 5.1.1.1

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

$x^{2} x^{1} + x^{2} \cdot 3 + 9 x + 9 \cdot 3$

Step 5.1.1.2

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

$x^{2 + 1} + x^{2} \cdot 3 + 9 x + 9 \cdot 3$

Step 5.1.2

Add $2$ and $1$ .

$x^{3} + x^{2} \cdot 3 + 9 x + 9 \cdot 3$

Step 5.2

Move $3$ to the left of $x^{2}$ .

$x^{3} + 3 \cdot x^{2} + 9 x + 9 \cdot 3$

Step 5.3

Multiply $9$ by $3$ .

$x^{3} + 3 x^{2} + 9 x + 27$