Finite Math Examples

$f (x) = x^{6} - 64$

Step 1

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$f (x) = {(x^{2})}^{3} - 64$

Step 2

Rewrite $64$ as $4^{3}$ .

$f (x) = {(x^{2})}^{3} - 4^{3}$

Step 3

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 $a = x^{2}$ and $b = 4$ .

$f (x) = (x^{2} - 4) ({(x^{2})}^{2} + x^{2} \cdot 4 + 4^{2})$

Step 4

Simplify.

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Step 4.1

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

$f (x) = (x^{2} - 2^{2}) ({(x^{2})}^{2} + x^{2} \cdot 4 + 4^{2})$

Step 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 $b = 2$ .

$f (x) = (x + 2) (x - 2) ({(x^{2})}^{2} + x^{2} \cdot 4 + 4^{2})$

Step 4.3

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 4.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (x) = (x + 2) (x - 2) (x^{2 \cdot 2} + x^{2} \cdot 4 + 4^{2})$

Step 4.3.2

Multiply $2$ by $2$ .

$f (x) = (x + 2) (x - 2) (x^{4} + x^{2} \cdot 4 + 4^{2})$

Step 4.4

Move $4$ to the left of $x^{2}$ .

$f (x) = (x + 2) (x - 2) (x^{4} + 4 x^{2} + 4^{2})$

Step 4.5

Raise $4$ to the power of $2$ .

$f (x) = (x + 2) (x - 2) (x^{4} + 4 x^{2} + 16)$

Step 4.6

Factor.

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Step 4.6.1

Rewrite $x^{4} + 4 x^{2} + 16$ in a factored form.

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Step 4.6.1.1

Rewrite the middle term.

$f (x) = (x + 2) (x - 2) (x^{4} + 2 x^{2} \cdot 4 - 4 x^{2} + 16)$

Step 4.6.1.2

Rearrange terms.

$f (x) = (x + 2) (x - 2) (x^{4} + 2 x^{2} \cdot 4 + 16 - 4 x^{2})$

Step 4.6.1.3

Factor first three terms by perfect square rule.

$f (x) = (x + 2) (x - 2) ({(x^{2} + 4)}^{2} - 4 x^{2})$

Step 4.6.1.4

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

$f (x) = (x + 2) (x - 2) ({(x^{2} + 4)}^{2} - {(2 x)}^{2})$

Step 4.6.1.5

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

$f (x) = (x + 2) (x - 2) ((x^{2} + 4 + 2 x) (x^{2} + 4 - (2 x)))$

Step 4.6.1.6

Simplify.

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Step 4.6.1.6.1

Reorder terms.

$f (x) = (x + 2) (x - 2) ((x^{2} + 2 x + 4) (x^{2} + 4 - (2 x)))$

Step 4.6.1.6.2

Multiply $2$ by $- 1$ .

$f (x) = (x + 2) (x - 2) ((x^{2} + 2 x + 4) (x^{2} + 4 - 2 x))$

Step 4.6.1.6.3

Reorder terms.

$f (x) = (x + 2) (x - 2) ((x^{2} + 2 x + 4) (x^{2} - 2 x + 4))$

Step 4.6.2

Remove unnecessary parentheses.

$f (x) = (x + 2) (x - 2) (x^{2} + 2 x + 4) (x^{2} - 2 x + 4)$