To find the least common denominator (LCD) of the expression
\[ \frac{t-5}{1-t} - \frac{t+4}{t+1} + \frac{t+2}{t^2-1}, \]
we first need to factor any denominators that can be factored.
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The first denominator is \(1 - t\), which can be rewritten as \(-(t - 1)\). Hence, we can write it as: \[ 1 - t = -(t - 1). \]
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The second denominator is \(t + 1\), which is already in simplest form.
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The third denominator is \(t^2 - 1\), which is a difference of squares. It factors as: \[ t^2 - 1 = (t - 1)(t + 1). \]
Now we have the following denominators:
- \(1 - t = -(t - 1)\)
- \(t + 1\)
- \(t^2 - 1 = (t - 1)(t + 1)\)
Next, we determine the least common denominator by taking the highest power of each factor present in any of the denominators:
- From \(1 - t\) (or \(t - 1\)): \(t - 1\)
- From \(t + 1\): \(t + 1\)
Thus, the LCD will be the product of all unique factors:
\[ \text{LCD} = (t - 1)(t + 1). \]
So, the least common denominator of the given expression is
\[ (t - 1)(t + 1). \]