What is the LCD in its factored form of the following rational expression subtraction?

Question

x2+2xx2+7x−8−xx−1

(1 point)
Responses

(x−1)(x2+7x−8)
left parenthesis x minus 1 right parenthesis left parenthesis x squared plus 7 x minus 8 right parenthesis

(x−1)2(x+8)
left parenthesis x minus 1 right parenthesis squared left parenthesis x plus 8 right parenthesis

x(x+2)
x left parenthesis x plus 2 right parenthesis

(x−1)(x+8)

Answer 1

To find the least common denominator (LCD) of the given rational expressions in the subtraction \( \frac{x^2 + 2x}{x^2 + 7x - 8} - \frac{x}{x - 1} \), we need to factor each denominator.

1. Factor \( x^2 + 7x - 8 \):

   • We need to find two numbers that multiply to \(-8\) and add to \(7\). These numbers are \(8\) and \(-1\).
   • Therefore, we can factor it as: \[ x^2 + 7x - 8 = (x + 8)(x - 1) \]

2. The second denominator: \( x - 1 \) is already factored.

Now, we can determine the LCD. The denominators we have after factoring are:

• \( (x + 8)(x - 1) \) from \( \frac{x^2 + 2x}{x^2 + 7x - 8} \)
• \( x - 1 \) from \( \frac{x}{x - 1} \)

The LCD will be:

• The highest power of each factor present in the denominators. Here, we have:
  • The factor \( x + 8 \) once.
  • The factor \( x - 1 \) once, since it's common in both and doesn't need a higher power.

Thus, the LCD in its factored form will be: \[ (x - 1)(x + 8) \]

Therefore, the correct response is: \((x - 1)(x + 8)\)