Find the least common denominator.

StartFraction 1 Over x squared minus 1 EndFraction1x2−1 

and 

StartFraction 1 Over x left parenthesis x minus 1 right parenthesis EndFraction1x(x−1) 

To find the least common denominator (LCD) of the two fractions, we need to find the least common multiple (LCM) of the denominators of both fractions.

### Step 1: Factorize the denominators
- The first denominator is \( x^2 - 1 \), which is a difference of squares. It can be factored as:
  \[
  x^2 - 1 = (x - 1)(x + 1)
  \]
- The second denominator is \( x(x - 1) \), which is already factored.

### Step 2: Identify the unique factors
The denominators after factorization are:
- \( (x - 1)(x + 1) \) from the first fraction.
- \( x(x - 1) \) from the second fraction.

### Step 3: Find the Least Common Denominator
The least common denominator is the product of all unique factors, each taken at the highest power that appears in the factorizations:

- The factors are \( x \), \( (x - 1) \), and \( (x + 1) \).
- The LCD is:
  \[
  \text{LCD} = x(x - 1)(x + 1)
  \]

So, the least common denominator is \( x(x - 1)(x + 1) \).

Would you like further details or have any other questions?

Here are some related questions:
1. How do you find the LCD for three or more fractions?
2. What happens if one of the factors in the denominator is a quadratic expression?
3. How can the LCD be used to add or subtract fractions?
4. How does the LCD help in solving equations with fractions?
5. What are some common mistakes when finding the LCD?

**Tip:** Always factorize completely before finding the least common denominator.

Find the least common denominator.

StartFraction 1 Over x squared minus 1 EndFraction1x2−1

and

StartFraction 1 Over x left parenthesis x minus 1 right parenthesis EndFraction1x(x−1)

Learn how to find the least common denominator for the rational expressions 1/(x^2 - 1) and 1/x(x - 1). This step-by-step solution includes factoring techniques such as the difference of squares and determining the LCM of denominators. Ideal for students in Grades 7-9.

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