Algebra Examples

$y x^{6} + 27 y^{7}$

Step 1

Factor $y$ out of $y x^{6} + 27 y^{7}$ .

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Step 1.1

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

$y (x^{6}) + 27 y^{7}$

Step 1.2

Factor $y$ out of $27 y^{7}$ .

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

Step 1.3

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

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

Step 2

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

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

Step 3

Rewrite $27 y^{6}$ as ${(3 y^{2})}^{3}$ .

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

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 $b = 3 y^{2}$ .

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.1.1.2

Multiply $2$ by $2$ .

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

Step 5.1.2

Rewrite using the commutative property of multiplication.

$y ((x^{2} + 3 y^{2}) (x^{4} - 1 \cdot 3 x^{2} y^{2} + {(3 y^{2})}^{2}))$

Step 5.1.3

Multiply $- 1$ by $3$ .

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

Step 5.1.4

Apply the product rule to $3 y^{2}$ .

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

Step 5.1.5

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

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

Step 5.1.6

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

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Step 5.1.6.1

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

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

Step 5.1.6.2

Multiply $2$ by $2$ .

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

Step 5.1.7

Factor.

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Step 5.1.7.1

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

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Step 5.1.7.1.1

Rewrite the middle term.

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

Step 5.1.7.1.2

Rearrange terms.

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

Step 5.1.7.1.3

Factor first three terms by perfect square rule.

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

Step 5.1.7.1.4

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

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

Step 5.1.7.1.5

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

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

Step 5.1.7.1.6

Simplify.

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Step 5.1.7.1.6.1

Multiply $3$ by $- 1$ .

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

Step 5.1.7.1.6.2

Remove parentheses.

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

Step 5.1.7.2

Remove unnecessary parentheses.

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

Step 5.2

Remove unnecessary parentheses.

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