To find the Least Common Denominator (LCD) of the rational expressions given, we first need to factor the denominators of the expressions. The expressions given are:
- \(\frac{x^2 + 2x}{x^2 + 7x - 8}\)
- \(\frac{x}{x - 1}\)
Step 1: Factor the denominators.
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For \(x^2 + 7x - 8\): We need to factor this quadratic expression. We will look for two numbers that multiply to \(-8\) and add to \(7\). The numbers are \(8\) and \(-1\).
Therefore: \[ x^2 + 7x - 8 = (x + 8)(x - 1) \]
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The second denominator is already factored: \(x - 1\)
Step 2: Identify the unique factors.
- From \((x + 8)(x - 1)\), we have the factors \(x + 8\) and \(x - 1\).
- From the second denominator \(x - 1\), we only have \(x - 1\).
Step 3: Write the LCD as the product of the unique factors. The unique factors from both denominators are \(x + 8\) and \(x - 1\).
Thus, the Least Common Denominator (LCD) in its factored form is: \[ \boxed{(x + 8)(x - 1)} \]
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