Algebra Examples

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

Step 1

Regroup terms.

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

Step 2

Factor $- 1$ out of $- x^{3} - y^{3}$ .

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Step 2.1

Factor $- 1$ out of $- x^{3}$ .

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

Step 2.2

Factor $- 1$ out of $- y^{3}$ .

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

Step 2.3

Factor $- 1$ out of $- (x^{3}) - (y^{3})$ .

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

Step 3

Factor.

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Step 3.1

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 $a = x$ and $b = y$ .

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

Step 3.2

Remove unnecessary parentheses.

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

Step 4

Factor out the greatest common factor from each group.

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Step 4.1

Group the first two terms and the last two terms.

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

Step 4.2

Factor out the greatest common factor (GCF) from each group.

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

Step 5

Factor the polynomial by factoring out the greatest common factor, $x + y$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Apply the distributive property.

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

Step 8

Simplify.

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Step 8.1

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

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

Step 8.2

Multiply $- 1 (- x y)$ .

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Step 8.2.1

Multiply $- 1$ by $- 1$ .

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

Step 8.2.2

Multiply $x$ by $1$ .

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

Step 8.3

Rewrite $- 1 y^{2}$ as $- y^{2}$ .

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

Step 9

Add $x y$ and $x y$ .

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

Step 10

Factor.

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Step 10.1

Rewrite $- x^{2} + 2 x y - y^{2} + 1$ in a factored form.

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Step 10.1.1

Rewrite $1$ as $1^{2}$ .

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

Step 10.1.2

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

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Step 10.1.2.1

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

$2 x y = 2 \cdot x \cdot y$

Step 10.1.2.2

Rewrite the polynomial.

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

Step 10.1.2.3

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

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

Step 10.1.3

Reorder $- {(x - y)}^{2}$ and $1^{2}$ .

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

Step 10.1.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x - y$ .

$(x + y) ((1 + x - y) (1 - (x - y)))$

Step 10.1.5

Simplify.

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Step 10.1.5.1

Apply the distributive property.

$(x + y) ((1 + x - y) (1 - x - - y))$

Step 10.1.5.2

Multiply $- - y$ .

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Step 10.1.5.2.1

Multiply $- 1$ by $- 1$ .

$(x + y) ((1 + x - y) (1 - x + 1 y))$

Step 10.1.5.2.2

Multiply $y$ by $1$ .

$(x + y) ((1 + x - y) (1 - x + y))$

Step 10.2

Remove unnecessary parentheses.

$(x + y) (1 + x - y) (1 - x + y)$

Step 11

Factor.

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Step 11.1

Factor $- 1$ out of $1 - x + y$ .

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Step 11.1.1

Move $1$ .

$(x + y) (1 + x - y) (- x + y + 1)$

Step 11.1.2

Factor $- 1$ out of $- x$ .

$(x + y) (1 + x - y) (- (x) + y + 1)$

Step 11.1.3

Factor $- 1$ out of $y$ .

$(x + y) (1 + x - y) (- (x) - 1 (- y) + 1)$

Step 11.1.4

Rewrite $1$ as $- 1 (- 1)$ .

$(x + y) (1 + x - y) (- (x) - 1 (- y) - 1 (- 1))$

Step 11.1.5

Factor $- 1$ out of $- (x) - 1 (- y)$ .

$(x + y) (1 + x - y) (- (x - y) - 1 (- 1))$

Step 11.1.6

Factor $- 1$ out of $- (x - y) - 1 (- 1)$ .

$(x + y) (1 + x - y) (- (x - y - 1))$

Step 11.2

Remove unnecessary parentheses.

$(x + y) (1 + x - y) \cdot - 1 (x - y - 1)$

Step 12

Factor out negative.

$- ((x + y) (1 + x - y) (x - y - 1))$

Step 13

Remove unnecessary parentheses.

$- (x + y) (1 + x - y) (x - y - 1)$