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

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

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Step 4.1.1

Move $x^{2}$ .

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.3

Add $2$ and $1$ .

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

Step 4.2

Rewrite using the commutative property of multiplication.

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

Step 4.3

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

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Step 4.3.1

Move $y$ .

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

Step 4.3.2

Multiply $y$ by $y$ .

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

Step 4.4

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

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Step 4.4.1

Move $x$ .

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

Step 4.4.2

Multiply $x$ by $x$ .

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

Step 4.5

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

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Step 4.5.1

Move $y^{2}$ .

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

Step 4.5.2

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

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Step 4.5.2.1

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

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

Step 4.5.2.2

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

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

Step 4.5.3

Add $2$ and $1$ .

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

Step 4.6

Rewrite $y x^{3} + 2 y^{2} x^{2} + y^{3} x$ in a factored form.

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Step 4.6.1

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

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Step 4.6.1.1

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 4.6.1.2

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 4.6.1.3

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 4.6.1.4

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 4.6.1.5

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 4.6.2

Factor using the perfect square rule.

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Step 4.6.2.1

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

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

Step 4.6.2.2

Rewrite the polynomial.

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

Step 4.6.2.3

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

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

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

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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