Question

(t-5)/(1-t)-(t+4)/(t+1)+(t+2)/(t^(2)-1). Find the least common denominater

Answer 1

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.

1. The first denominator is \(1 - t\), which can be rewritten as \(-(t - 1)\). Hence, we can write it as:
\[
1 - t = -(t - 1).
\]

2. The second denominator is \(t + 1\), which is already in simplest form.

3. 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

\[
(t - 1)(t + 1).
\]