Algebra Examples

$x^{6} + 6 x^{4} + 12 x^{2} + 8$

Step 1

Regroup terms.

$x^{6} + 8 + 6 x^{4} + 12 x^{2}$

Step 2

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

${(x^{2})}^{3} + 8 + 6 x^{4} + 12 x^{2}$

Step 3

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

${(x^{2})}^{3} + 2^{3} + 6 x^{4} + 12 x^{2}$

Step 4

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

$(x^{2} + 2) ({(x^{2})}^{2} - x^{2} \cdot 2 + 2^{2}) + 6 x^{4} + 12 x^{2}$

Step 5

Simplify.

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Step 5.1

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

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Step 5.1.1

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

$(x^{2} + 2) (x^{2 \cdot 2} - x^{2} \cdot 2 + 2^{2}) + 6 x^{4} + 12 x^{2}$

Step 5.1.2

Multiply $2$ by $2$ .

$(x^{2} + 2) (x^{4} - x^{2} \cdot 2 + 2^{2}) + 6 x^{4} + 12 x^{2}$

Step 5.2

Multiply $2$ by $- 1$ .

$(x^{2} + 2) (x^{4} - 2 x^{2} + 2^{2}) + 6 x^{4} + 12 x^{2}$

Step 5.3

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

$(x^{2} + 2) (x^{4} - 2 x^{2} + 4) + 6 x^{4} + 12 x^{2}$

Step 6

Factor $6 x^{2}$ out of $6 x^{4} + 12 x^{2}$ .

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Step 6.1

Factor $6 x^{2}$ out of $6 x^{4}$ .

$(x^{2} + 2) (x^{4} - 2 x^{2} + 4) + 6 x^{2} (x^{2}) + 12 x^{2}$

Step 6.2

Factor $6 x^{2}$ out of $12 x^{2}$ .

$(x^{2} + 2) (x^{4} - 2 x^{2} + 4) + 6 x^{2} (x^{2}) + 6 x^{2} (2)$

Step 6.3

Factor $6 x^{2}$ out of $6 x^{2} (x^{2}) + 6 x^{2} (2)$ .

$(x^{2} + 2) (x^{4} - 2 x^{2} + 4) + 6 x^{2} (x^{2} + 2)$

Step 7

Factor $x^{2} + 2$ out of $(x^{2} + 2) (x^{4} - 2 x^{2} + 4) + 6 x^{2} (x^{2} + 2)$ .

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Step 7.1

Factor $x^{2} + 2$ out of $6 x^{2} (x^{2} + 2)$ .

$(x^{2} + 2) (x^{4} - 2 x^{2} + 4) + (x^{2} + 2) (6 x^{2})$

Step 7.2

Factor $x^{2} + 2$ out of $(x^{2} + 2) (x^{4} - 2 x^{2} + 4) + (x^{2} + 2) (6 x^{2})$ .

$(x^{2} + 2) (x^{4} - 2 x^{2} + 4 + 6 x^{2})$

Step 8

Add $- 2 x^{2}$ and $6 x^{2}$ .

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

Step 9

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

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

Step 10

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

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

Step 11

Factor using the perfect square rule.

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Step 11.1

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

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

Step 11.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 u = 2 \cdot u \cdot 2$

Step 11.3

Rewrite the polynomial.

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

Step 11.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 2$ .

$(x^{2} + 2) {(u + 2)}^{2}$

Step 12

Replace all occurrences of $u$ with $x^{2}$ .

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