Question

What is the LCD in its factored form of the following rational expression subtraction?%0D%0A%0D%0Ax2+2xx2+7x−8−xx−1%0D%0A%0D%0A(1 point)%0D%0AResponses

Answer 1

To find the Least Common Denominator (LCD) of the rational expressions given, we first need to factor the denominators of the expressions. The expressions given are:

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

**Step 1:** Factor the denominators.

1. For \(x^2 + 7x - 8\):
We need to factor this quadratic expression. We will look for two numbers that multiply to \(-8\) and add to \(7\). The numbers are \(8\) and \(-1\).

Therefore:
\[
x^2 + 7x - 8 = (x + 8)(x - 1)
\]

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

**Step 2:** Identify the unique factors.
- From \((x + 8)(x - 1)\), we have the factors \(x + 8\) and \(x - 1\).
- From the second denominator \(x - 1\), we only have \(x - 1\).

**Step 3:** Write the LCD as the product of the unique factors.
The unique factors from both denominators are \(x + 8\) and \(x - 1\).

Thus, the **Least Common Denominator (LCD)** in its factored form is:
\[
\boxed{(x + 8)(x - 1)}
\]