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Question

Answer

$\frac{\frac{\frac{2}{x} + \frac{2}{x ^{2}}}{1}}{x} - x = \frac{- x ^{5} + 2 x ^{2} + 2 x}{x ^{4}}$

Explanation

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$\frac{\frac{\frac{2}{x} + \frac{2}{x ^{2}}}{1}}{x} - x \overset{◯}{1} \frac{\frac{\frac{2 x ^{2} + 2 x}{x ^{3}}}{1}}{x} - x \overset{◯}{2} \frac{\frac{2 x ^{2} + 2 x}{x ^{3}}}{x} - x \overset{◯}{3} \frac{2 x ^{2} + 2 x}{x ^{4}} - x \overset{◯}{4} \frac{- x ^{5} + 2 x ^{2} + 2 x}{x ^{4}}$
①	Add $\frac{2}{x}$ and $\frac{2}{x ^{2}}$ to get $\frac{2 x ^{2} + 2 x}{x ^{3}}$ . To add raitonal expressions, both fractions must have the same denominator. We can create a common denominator by multiplying the first fraction by $x^{2}$ and the second by $x$ . $\frac{2}{x} + \frac{2}{x ^{2}} = \frac{2 \cdot x ^{2}}{x \cdot x ^{2}} + \frac{2 \cdot x}{x ^{2} \cdot x} = = \frac{2 x ^{2}}{x ^{3}} + \frac{2 x}{x ^{3}} = \frac{2 x ^{2} + 2 x}{x ^{3}}$
②	Divide $\frac{2 x ^{2} + 2 x}{x ^{3}}$ by $1$ to get $\frac{2 x ^{2} + 2 x}{x ^{3}}$ . Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $\frac{\frac{2 x ^{2} + 2 x}{x ^{3}}}{1} Step 1 \frac{2 x ^{2} + 2 x}{x ^{3}} \cdot \frac{1}{1} Step 2 \frac{( 2 x ^{2} + 2 x ) \cdot 1}{x ^{3} \cdot 1} = Step 3 \frac{2 x ^{2} + 2 x}{x ^{3}}$
③	Divide $\frac{2 x ^{2} + 2 x}{x ^{3}}$ by $x$ to get $\frac{2 x ^{2} + 2 x}{x ^{4}}$ . Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $\frac{\frac{2 x ^{2} + 2 x}{x ^{3}}}{x} Step 1 \frac{2 x ^{2} + 2 x}{x ^{3}} \cdot \frac{1}{x} Step 2 \frac{( 2 x ^{2} + 2 x ) \cdot 1}{x ^{3} \cdot x} = Step 3 \frac{2 x ^{2} + 2 x}{x ^{4}}$
④	Subtract $x$ from $\frac{2 x ^{2} + 2 x}{x ^{4}}$ to get $\frac{- x ^{5} + 2 x ^{2} + 2 x}{x ^{4}}$ . Step 1: Write $x$ as a fraction by putting $1$ in the denominator. Step 2: To subtract raitonal expressions, both fractions must have the same denominator. We can create a common denominator by multiplying the second fraction by $x^{4}$ . $\frac{2 x ^{2} + 2 x}{x ^{4}} - x Step 1 \frac{2 x ^{2} + 2 x}{x ^{4}} - \frac{x}{1} = \frac{2 x ^{2} + 2 x}{x ^{4}} - \frac{x \cdot x ^{4}}{1 \cdot x ^{4}} = Step 2 \frac{2 x ^{2} + 2 x}{x ^{4}} - \frac{x ^{5}}{x ^{4}} = \frac{- x ^{5} + 2 x ^{2} + 2 x}{x ^{4}}$

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