Question
Find the least common denominator.
StartFraction 1 Over x squared minus 25 EndFraction
and StartFraction 3 Over x left parenthesis x minus 5 right parenthesis EndFraction
StartFraction 1 Over x squared minus 25 EndFraction
and StartFraction 3 Over x left parenthesis x minus 5 right parenthesis EndFraction
Answers
GPT-4o mini
To find the least common denominator (LCD) for the fractions \(\frac{1}{x^2 - 25}\) and \(\frac{3}{x(x - 5)}\), we first need to factor the denominators.
1. **Factor \(x^2 - 25\)**:
\[
x^2 - 25 \text{ is a difference of squares, which factors to } (x - 5)(x + 5).
\]
2. **Identify the denominators**:
- The first fraction has a denominator of \( (x - 5)(x + 5) \).
- The second fraction has a denominator of \( x(x - 5) \).
3. **Determine the LCD**:
- The unique factors in the denominators are \(x\), \(x - 5\), and \(x + 5\).
- The least common denominator is found by taking each factor to its highest power:
\[
\text{LCD} = x(x - 5)(x + 5).
\]
Thus, the least common denominator of the fractions \(\frac{1}{x^2 - 25}\) and \(\frac{3}{x(x - 5)}\) is:
\[
\boxed{x(x - 5)(x + 5)}.
\]
1. **Factor \(x^2 - 25\)**:
\[
x^2 - 25 \text{ is a difference of squares, which factors to } (x - 5)(x + 5).
\]
2. **Identify the denominators**:
- The first fraction has a denominator of \( (x - 5)(x + 5) \).
- The second fraction has a denominator of \( x(x - 5) \).
3. **Determine the LCD**:
- The unique factors in the denominators are \(x\), \(x - 5\), and \(x + 5\).
- The least common denominator is found by taking each factor to its highest power:
\[
\text{LCD} = x(x - 5)(x + 5).
\]
Thus, the least common denominator of the fractions \(\frac{1}{x^2 - 25}\) and \(\frac{3}{x(x - 5)}\) is:
\[
\boxed{x(x - 5)(x + 5)}.
\]