Question

what is the lcd in its factored form of the following rational expression subtraction?\\(\frac{x^{2}+2x}{x^{2}+7x - 8}-\frac{x}{x - 1}\\)\\((1\text{ point})\\)\\(circ (x - 1)(x + 8)\\)\\(circ (x - 1)(x^{2}+7x - 8)\\)\\(circ x(x + 2)\\)\\(circ (x - 1)^{2}(x + 8)\\)

Explanation:

Step1: Factor the denominator of the first fraction

The denominator of the first rational expression is \(x^{2}+7x - 8\). We factor it by finding two numbers that multiply to \(- 8\) and add to \(7\). The numbers are \(8\) and \(-1\). So, \(x^{2}+7x - 8=(x + 8)(x - 1)\).

Step2: Identify the denominators

The denominators of the two rational expressions are \((x + 8)(x - 1)\) (from the first fraction) and \(x - 1\) (from the second fraction).

Step3: Determine the LCD

The Least Common Denominator (LCD) is the product of the highest power of each distinct factor. Here, the distinct factors are \((x - 1)\) and \((x + 8)\). The highest power of \((x - 1)\) is \(1\) (since it appears once in the factored form of the first denominator and once in the second denominator) and the highest power of \((x + 8)\) is \(1\). So, the LCD is \((x - 1)(x + 8)\).

Answer:

\((x - 1)(x + 8)\) (corresponding to the first option: \((x - 1)(x + 8)\))