Precalculus Examples

${(x + h)}^{3} - 8$

Step 1

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

${(x + h)}^{3} - 2^{3}$

Step 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 $a = x + h$ and $b = 2$ .

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

Apply the distributive property.

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

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $x$ by $x$ .

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

Step 3.3.1.2

Multiply $h$ by $h$ .

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

Step 3.3.2

Add $x h$ and $h x$ .

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Step 3.3.2.1

Reorder $x$ and $h$ .

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

Step 3.3.2.2

Add $h x$ and $h x$ .

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

Step 3.4

Apply the distributive property.

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

Step 3.5

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

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

Step 3.6

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

$(x + h - 2) (x^{2} + 2 h x + h^{2} + 2 x + 2 h + 2^{2})$

Step 3.7

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

$(x + h - 2) (x^{2} + 2 h x + h^{2} + 2 x + 2 h + 4)$