Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

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

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

Step 1.1.2

Factor using the perfect square rule.

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Step 1.1.2.1

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 1.1.2.2

Rewrite the polynomial.

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

Step 1.1.2.3

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

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

Step 1.2

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

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

Step 1.3

Cancel the common factor of $x - y$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

Cancel the common factor of $x^{2} + x y + y^{2}$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 1.5

Cancel the common factor of ${(x + y)}^{2}$ and $x + y$ .

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Step 1.5.1

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

$\frac{(x + y) (x + y)}{x + y} - x - y$

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Multiply by $1$ .

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

Step 1.5.2.2

Cancel the common factor.

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

Step 1.5.2.3

Rewrite the expression.

$\frac{x + y}{1} - x - y$

Step 1.5.2.4

Divide $x + y$ by $1$ .

$x + y - x - y$

Step 2

Combine the opposite terms in $x + y - x - y$ .

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Step 2.1

Subtract $x$ from $x$ .

$y + 0 - y$

Step 2.2

Add $y$ and $0$ .

$y - y$

Step 2.3

Subtract $y$ from $y$ .

$0$