Algebra Examples

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

Step 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 $b = y$ .

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

Step 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 $b = y$ .

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x + y$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Raise $x - y$ to the power of $1$ .

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

Step 5

Raise $x - y$ to the power of $1$ .

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

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7

Simplify the expression.

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Step 7.1

Add $1$ and $1$ .

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

Step 7.2

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

$(x - y) (x - y)$

Step 8

Expand $(x - y) (x - y)$ using the FOIL Method.

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Step 8.1

Apply the distributive property.

$x (x - y) - y (x - y)$

Step 8.2

Apply the distributive property.

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

Step 8.3

Apply the distributive property.

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

Step 9

Simplify and combine like terms.

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Step 9.1

Simplify each term.

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Step 9.1.1

Multiply $x$ by $x$ .

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

Step 9.1.2

Rewrite using the commutative property of multiplication.

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

Step 9.1.3

Rewrite using the commutative property of multiplication.

$x^{2} - x y - y x - 1 \cdot - 1 y \cdot y$

Step 9.1.4

Multiply $y$ by $y$ by adding the exponents.

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Step 9.1.4.1

Move $y$ .

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

Step 9.1.4.2

Multiply $y$ by $y$ .

$x^{2} - x y - y x - 1 \cdot - 1 y^{2}$

Step 9.1.5

Multiply $- 1$ by $- 1$ .

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

Step 9.1.6

Multiply $y^{2}$ by $1$ .

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

Step 9.2

Subtract $y x$ from $- x y$ .

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Step 9.2.1

Move $y$ .

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

Step 9.2.2

Subtract $x y$ from $- x y$ .

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