Algebra Examples

$3 x^{6} y^{2} + 81 y^{2}$

Step 1

Factor $3 y^{2}$ out of $3 x^{6} y^{2} + 81 y^{2}$ .

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Step 1.1

Factor $3 y^{2}$ out of $3 x^{6} y^{2}$ .

$3 y^{2} (x^{6}) + 81 y^{2}$

Step 1.2

Factor $3 y^{2}$ out of $81 y^{2}$ .

$3 y^{2} (x^{6}) + 3 y^{2} (27)$

Step 1.3

Factor $3 y^{2}$ out of $3 y^{2} (x^{6}) + 3 y^{2} (27)$ .

$3 y^{2} (x^{6} + 27)$

Step 2

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

$3 y^{2} ({(x^{2})}^{3} + 27)$

Step 3

Rewrite $27$ as $3^{3}$ .

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

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 $b = 3$ .

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

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

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

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Step 5.1.1.1

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

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

Step 5.1.1.2

Multiply $2$ by $2$ .

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

Step 5.1.2

Multiply $3$ by $- 1$ .

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

Step 5.1.3

Raise $3$ to the power of $2$ .

$3 y^{2} ((x^{2} + 3) (x^{4} - 3 x^{2} + 9))$

Step 5.1.4

Factor.

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Step 5.1.4.1

Rewrite $x^{4} - 3 x^{2} + 9$ in a factored form.

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Step 5.1.4.1.1

Rewrite the middle term.

$3 y^{2} ((x^{2} + 3) (x^{4} + 2 x^{2} \cdot 3 - 9 x^{2} + 9))$

Step 5.1.4.1.2

Rearrange terms.

$3 y^{2} ((x^{2} + 3) (x^{4} + 2 x^{2} \cdot 3 + 9 - 9 x^{2}))$

Step 5.1.4.1.3

Factor first three terms by perfect square rule.

$3 y^{2} ((x^{2} + 3) ({(x^{2} + 3)}^{2} - 9 x^{2}))$

Step 5.1.4.1.4

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

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

Step 5.1.4.1.5

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

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

Step 5.1.4.1.6

Simplify.

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Step 5.1.4.1.6.1

Reorder terms.

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

Step 5.1.4.1.6.2

Multiply $3$ by $- 1$ .

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

Step 5.1.4.1.6.3

Reorder terms.

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

Step 5.1.4.2

Remove unnecessary parentheses.

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

Step 5.2

Remove unnecessary parentheses.

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