Algebra Examples

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

Step 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 $b = y$ .

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

Step 2

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 $b = y$ .

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

Step 3

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 $b = y$ .

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

Step 4

Factor using the perfect square rule.

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Step 4.1

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 4.2

Rewrite the polynomial.

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

Step 4.3

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

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

Step 5

Simplify terms.

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Step 5.1

Combine.

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

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

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

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

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

Step 5.3

Cancel the common factor of $x + y$ .

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Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

Rewrite the expression.

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

Step 5.4

Cancel the common factor of $x - y$ .

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Step 5.4.1

Cancel the common factor.

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

Step 5.4.2

Rewrite the expression.

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