Question
What is the LCD in its factored from of the following rational expression subtraction?
x2+2x/x2+7x-8 - x/x-1
x2+2x/x2+7x-8 - x/x-1
Answers
GPT-4o mini
To find the least common denominator (LCD) of the rational expressions in the subtraction \( \frac{x^2 + 2x}{x^2 + 7x - 8} - \frac{x}{x - 1} \), we need to factor the denominators of both fractions.
1. **Factor the denominator of the first fraction:**
The denominator is \( x^2 + 7x - 8 \). We can factor this quadratic expression by looking for two numbers that multiply to \(-8\) (the constant term) and add to \(7\) (the coefficient of \(x\)):
\[
x^2 + 7x - 8 = (x + 8)(x - 1)
\]
2. **Factor the denominator of the second fraction:**
The denominator is \( x - 1 \), which is already in factored form.
3. **Determine the LCD:**
The LCD must include all factors present in the denominators, at their highest powers. Thus, we take:
- From \( (x + 8)(x - 1) \), we have \( (x + 8) \) and \( (x - 1) \).
- From \( x - 1 \), we already include \( (x - 1) \).
Therefore, the LCD is:
\[
\text{LCD} = (x + 8)(x - 1)
\]
In summary, the least common denominator of the given expression, in its factored form, is:
\[
\boxed{(x + 8)(x - 1)}
\]
1. **Factor the denominator of the first fraction:**
The denominator is \( x^2 + 7x - 8 \). We can factor this quadratic expression by looking for two numbers that multiply to \(-8\) (the constant term) and add to \(7\) (the coefficient of \(x\)):
\[
x^2 + 7x - 8 = (x + 8)(x - 1)
\]
2. **Factor the denominator of the second fraction:**
The denominator is \( x - 1 \), which is already in factored form.
3. **Determine the LCD:**
The LCD must include all factors present in the denominators, at their highest powers. Thus, we take:
- From \( (x + 8)(x - 1) \), we have \( (x + 8) \) and \( (x - 1) \).
- From \( x - 1 \), we already include \( (x - 1) \).
Therefore, the LCD is:
\[
\text{LCD} = (x + 8)(x - 1)
\]
In summary, the least common denominator of the given expression, in its factored form, is:
\[
\boxed{(x + 8)(x - 1)}
\]