Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $y x$ .

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Step 1.1

Multiply $\frac{x^{2} - 2 x y + y^{2}}{\frac{x}{y} - \frac{y}{x}}$ by $\frac{y x}{y x}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $y$ .

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Step 3.1.1

Factor $y$ out of $y x$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Cancel the common factor of $x$ .

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Step 3.6.1

Move the leading negative in $- \frac{y}{x}$ into the numerator.

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

Step 3.6.2

Factor $x$ out of $y x$ .

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

Step 3.6.3

Cancel the common factor.

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

Step 3.6.4

Rewrite the expression.

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Add $1$ and $1$ .

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

Step 4

Simplify the numerator.

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Step 4.1

Factor by grouping.

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Step 4.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 1 \cdot - 2 = - 2$ and whose sum is $b = 1$ .

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Step 4.1.1.1

Reorder terms.

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

Step 4.1.1.2

Reorder $- 2 x \cdot x y \cdot y$ and $y^{2} x y$ .

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

Step 4.1.1.3

Factor $1$ out of $y^{2} x y$ .

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

Step 4.1.1.4

Rewrite $1$ as $- 1$ plus $2$

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

Step 4.1.1.5

Apply the distributive property.

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

Step 4.1.1.6

Move parentheses.

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

Step 4.1.2

Combine exponents.

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.2.4

Add $1$ and $1$ .

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

Step 4.1.2.5

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

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

Step 4.1.2.6

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

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

Step 4.1.2.7

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

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

Step 4.1.2.8

Add $1$ and $1$ .

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

Step 4.1.3

Add $- 1 y^{2} x y$ and $2 y^{2} x y$ .

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

Step 4.1.4

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

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Step 4.1.4.1

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

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

Step 4.1.4.2

Factor $x^{2} y$ out of $1 y^{2} x y$ .

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

Step 4.1.4.3

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

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

Step 4.1.4.4

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

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

Step 4.1.4.5

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

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

Step 4.1.5

Combine exponents.

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Step 4.1.5.1

Multiply $y$ by $1$ .

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

Step 4.1.5.2

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

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

Step 4.1.5.3

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

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

Step 4.1.5.4

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

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

Step 4.1.5.5

Add $1$ and $1$ .

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

Step 4.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.3

Combine $\frac{1}{x}$ and $y^{2}$ .

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

Step 4.4

To write $x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

To write $- 2 y$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 4.7

Combine $- 2 y$ and $\frac{x}{x}$ .

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

Step 4.8

Combine the numerators over the common denominator.

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

Step 4.9

Factor using the perfect square rule.

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Step 4.9.1

Rearrange terms.

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

Step 4.9.2

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

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

Step 4.9.3

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

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

Step 4.9.4

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

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

Step 4.9.5

Add $1$ and $1$ .

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

Step 4.9.6

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

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

Step 4.9.7

Rewrite the polynomial.

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

Step 4.9.8

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

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

Step 4.10

Combine exponents.

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Step 4.10.1

Combine $x^{2}$ and $\frac{{(x - y)}^{2}}{x}$ .

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

Step 4.10.2

Combine $y$ and $\frac{x^{2} {(x - y)}^{2}}{x}$ .

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

Step 4.11

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

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Step 4.11.1

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

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

Step 4.11.2

Cancel the common factors.

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Step 4.11.2.1

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

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

Step 4.11.2.2

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

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

Step 4.11.2.3

Cancel the common factor.

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

Step 4.11.2.4

Rewrite the expression.

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

Step 4.11.2.5

Divide $y (x {(x - y)}^{2})$ by $1$ .

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

Step 4.12

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

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

Step 4.13

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

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Step 4.13.1

Apply the distributive property.

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

Step 4.13.2

Apply the distributive property.

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

Step 4.13.3

Apply the distributive property.

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

Step 4.14

Simplify and combine like terms.

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Step 4.14.1

Simplify each term.

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Step 4.14.1.1

Multiply $x$ by $x$ .

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

Step 4.14.1.2

Rewrite using the commutative property of multiplication.

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

Step 4.14.1.3

Rewrite using the commutative property of multiplication.

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

Step 4.14.1.4

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

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Step 4.14.1.4.1

Move $y$ .

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

Step 4.14.1.4.2

Multiply $y$ by $y$ .

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

Step 4.14.1.5

Multiply $- 1$ by $- 1$ .

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

Step 4.14.1.6

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

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

Step 4.14.2

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

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Step 4.14.2.1

Move $y$ .

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

Step 4.14.2.2

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

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

Step 4.15

Apply the distributive property.

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

Step 4.16

Simplify.

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Step 4.16.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 4.16.1.1

Multiply $x$ by $x^{2}$ .

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Step 4.16.1.1.1

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

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

Step 4.16.1.1.2

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

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

Step 4.16.1.2

Add $1$ and $2$ .

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

Step 4.16.2

Rewrite using the commutative property of multiplication.

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

Step 4.17

Multiply $x$ by $x$ by adding the exponents.

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Step 4.17.1

Move $x$ .

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

Step 4.17.2

Multiply $x$ by $x$ .

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

Step 4.18

Apply the distributive property.

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

Step 4.19

Simplify.

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Step 4.19.1

Rewrite using the commutative property of multiplication.

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

Step 4.19.2

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 4.19.2.1

Move $y^{2}$ .

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

Step 4.19.2.2

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

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Step 4.19.2.2.1

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

$\frac{y x^{3} - 2 y (x^{2} y) + y^{2} y^{1} x}{x^{2} - y^{2}}$

Step 4.19.2.2.2

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

$\frac{y x^{3} - 2 y (x^{2} y) + y^{2 + 1} x}{x^{2} - y^{2}}$

Step 4.19.2.3

Add $2$ and $1$ .

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

Step 4.20

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

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Step 4.20.1

Move $y$ .

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

Step 4.20.2

Multiply $y$ by $y$ .

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

Step 4.21

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

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Step 4.21.1

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

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

Step 4.21.2

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

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

Step 4.21.3

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

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

Step 4.21.4

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

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

Step 4.21.5

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

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

Step 4.22

Factor using the perfect square rule.

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Step 4.22.1

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

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

Step 4.22.2

Rewrite the polynomial.

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

Step 4.22.3

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

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

Step 5

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 $b = y$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

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

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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