Algebra Examples

$x^{5} + x^{3} y^{2} + x y^{4} + x^{4} + x^{2} y^{2} + y^{4}$

Step 1

Regroup terms.

$x^{5} + x^{3} y^{2} + x y^{4} + x^{4} + x^{2} y^{2} + y^{4}$

Step 2

Factor $x$ out of $x^{5} + x^{3} y^{2} + x y^{4}$ .

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Step 2.1

Factor $x$ out of $x^{5}$ .

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

Step 2.2

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

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

Step 2.3

Factor $x$ out of $x y^{4}$ .

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

Step 2.4

Factor $x$ out of $x \cdot x^{4} + x (x^{2} y^{2})$ .

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

Step 2.5

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

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

Step 3

Rewrite the middle term.

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

Step 4

Rearrange terms.

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

Step 5

Factor first three terms by perfect square rule.

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

Step 6

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

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

Step 7

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

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

Step 8

Factor.

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Step 8.1

Remove parentheses.

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

Step 8.2

Remove unnecessary parentheses.

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

Step 9

Rewrite the middle term.

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

Step 10

Rearrange terms.

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

Step 11

Factor first three terms by perfect square rule.

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

Step 12

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

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

Step 13

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

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

Step 14

Remove parentheses.

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

Step 15

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

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Step 15.1

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

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

Step 15.2

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

$(x^{2} + y^{2} + x y) (x^{2} + y^{2} - x y) (x) + (x^{2} + y^{2} + x y) (x^{2} + y^{2} - x y) (1)$

Step 15.3

Factor $(x^{2} + y^{2} + x y) (x^{2} + y^{2} - x y)$ out of $(x^{2} + y^{2} + x y) (x^{2} + y^{2} - x y) (x) + (x^{2} + y^{2} + x y) (x^{2} + y^{2} - x y) (1)$ .

$(x^{2} + y^{2} + x y) (x^{2} + y^{2} - x y) (x + 1)$