Algebra Examples

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

Step 1

Regroup terms.

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

Step 2

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

$2^{3} + a^{6} + 12 a^{2} + 6 a^{4}$

Step 3

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

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

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Multiply $2$ by $2$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 8

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

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

Step 9

Reorder terms.

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

Factor using the perfect square rule.

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Step 10.3.1

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

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

Step 10.3.2

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 10.3.3

Rewrite the polynomial.

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

Step 10.3.4

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

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

Step 10.4

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

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