Questions: (3x^2+3x)/(x^3-x)

(3x^2+3x)/(x^3-x)
Transcript text: \(\frac{3 x^{2}+3 x}{x^{3}-x}\)
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Solution

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Solution Steps

To simplify the given expression, we can start by factoring both the numerator and the denominator. The numerator can be factored as \(3x(x + 1)\) and the denominator can be factored as \(x(x^2 - 1)\). Further, \(x^2 - 1\) can be factored as \((x - 1)(x + 1)\). After factoring, we can cancel out the common terms in the numerator and the denominator.

Step 1: Factor the Numerator and Denominator

We start with the expression

\[ \frac{3x^2 + 3x}{x^3 - x}. \]

The numerator can be factored as

\[ 3x(x + 1), \]

and the denominator can be factored as

\[ x(x^2 - 1) = x(x - 1)(x + 1). \]

Step 2: Simplify the Expression

Now, substituting the factored forms into the expression, we have:

\[ \frac{3x(x + 1)}{x(x - 1)(x + 1)}. \]

We can cancel the common term \((x + 1)\) from the numerator and the denominator (assuming \(x \neq -1\)):

\[ \frac{3x}{x(x - 1)}. \]

Step 3: Further Simplification

This simplifies to:

\[ \frac{3}{x - 1} \quad \text{(assuming \(x \neq 0\) and \(x \neq 1\))}. \]

Final Answer

Thus, the simplified expression is

\[ \boxed{\frac{3}{x - 1}}. \]

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