Calculus Examples

$\frac{{(2 + x)}^{3} - 8}{x}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $8$ as $2^{3}$ .

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

Step 1.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 2 + x$ and $b = 2$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Subtract $2$ from $2$ .

$\frac{(x + 0) ({(2 + x)}^{2} + (2 + x) \cdot 2 + 2^{2})}{x}$

Step 1.3.2

Add $x$ and $0$ .

$\frac{x ({(2 + x)}^{2} + (2 + x) \cdot 2 + 2^{2})}{x}$

Step 1.3.3

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

$\frac{x ((2 + x) (2 + x) + (2 + x) \cdot 2 + 2^{2})}{x}$

Step 1.3.4

Expand $(2 + x) (2 + x)$ using the FOIL Method.

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Step 1.3.4.1

Apply the distributive property.

$\frac{x (2 (2 + x) + x (2 + x) + (2 + x) \cdot 2 + 2^{2})}{x}$

Step 1.3.4.2

Apply the distributive property.

$\frac{x (2 \cdot 2 + 2 x + x (2 + x) + (2 + x) \cdot 2 + 2^{2})}{x}$

Step 1.3.4.3

Apply the distributive property.

$\frac{x (2 \cdot 2 + 2 x + x \cdot 2 + x \cdot x + (2 + x) \cdot 2 + 2^{2})}{x}$

Step 1.3.5

Simplify and combine like terms.

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Step 1.3.5.1

Simplify each term.

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Step 1.3.5.1.1

Multiply $2$ by $2$ .

$\frac{x (4 + 2 x + x \cdot 2 + x \cdot x + (2 + x) \cdot 2 + 2^{2})}{x}$

Step 1.3.5.1.2

Move $2$ to the left of $x$ .

$\frac{x (4 + 2 x + 2 \cdot x + x \cdot x + (2 + x) \cdot 2 + 2^{2})}{x}$

Step 1.3.5.1.3

Multiply $x$ by $x$ .

$\frac{x (4 + 2 x + 2 x + x^{2} + (2 + x) \cdot 2 + 2^{2})}{x}$

Step 1.3.5.2

Add $2 x$ and $2 x$ .

$\frac{x (4 + 4 x + x^{2} + (2 + x) \cdot 2 + 2^{2})}{x}$

Step 1.3.6

Apply the distributive property.

$\frac{x (4 + 4 x + x^{2} + 2 \cdot 2 + x \cdot 2 + 2^{2})}{x}$

Step 1.3.7

Multiply $2$ by $2$ .

$\frac{x (4 + 4 x + x^{2} + 4 + x \cdot 2 + 2^{2})}{x}$

Step 1.3.8

Move $2$ to the left of $x$ .

$\frac{x (4 + 4 x + x^{2} + 4 + 2 \cdot x + 2^{2})}{x}$

Step 1.3.9

Raise $2$ to the power of $2$ .

$\frac{x (4 + 4 x + x^{2} + 4 + 2 x + 4)}{x}$

Step 1.3.10

Add $4$ and $4$ .

$\frac{x (4 x + x^{2} + 8 + 2 x + 4)}{x}$

Step 1.3.11

Add $4 x$ and $2 x$ .

$\frac{x (6 x + x^{2} + 8 + 4)}{x}$

Step 1.3.12

Add $8$ and $4$ .

$\frac{x (6 x + x^{2} + 12)}{x}$

Step 1.3.13

Reorder terms.

$\frac{x (x^{2} + 6 x + 12)}{x}$

Step 2

Cancel the common factor of $x$ .

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Step 2.1

Cancel the common factor.

$\frac{x (x^{2} + 6 x + 12)}{x}$

Step 2.2

Divide $x^{2} + 6 x + 12$ by $1$ .

$x^{2} + 6 x + 12$