Finite Math Examples

$x^{6} + 72 x^{3} + 512$

Step 1

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

${(x^{3})}^{2} + 72 x^{3} + 512$

Step 2

Let $u = x^{3}$ . Substitute $u$ for all occurrences of $x^{3}$ .

$u^{2} + 72 u + 512$

Step 3

Factor $u^{2} + 72 u + 512$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $512$ and whose sum is $72$ .

$8, 64$

Step 3.2

Write the factored form using these integers.

$(u + 8) (u + 64)$

Step 4

Replace all occurrences of $u$ with $x^{3}$ .

$(x^{3} + 8) (x^{3} + 64)$

Step 5

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

$(x^{3} + 2^{3}) (x^{3} + 64)$

Step 6

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

$(x + 2) (x^{2} - x \cdot 2 + 2^{2}) (x^{3} + 64)$

Step 7

Simplify.

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Step 7.1

Multiply $2$ by $- 1$ .

$(x + 2) (x^{2} - 2 x + 2^{2}) (x^{3} + 64)$

Step 7.2

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

$(x + 2) (x^{2} - 2 x + 4) (x^{3} + 64)$

Step 8

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

$(x + 2) (x^{2} - 2 x + 4) (x^{3} + 4^{3})$

Step 9

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 4$ .

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

Step 10

Factor.

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Step 10.1

Simplify.

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Step 10.1.1

Multiply $4$ by $- 1$ .

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

Step 10.1.2

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

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

Step 10.2

Remove unnecessary parentheses.

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