Question

look at the expression below.
\\(\frac{4w - 5}{w^2 + w - 2} + \frac{w^2}{w^2 + 7w + 10}\\)
which of the following is the least common denominator for this expression?
\\(\bigcirc\\) a. \\((w + 5)(w - 1)(w + 2)\\)
\\(\bigcirc\\) b. \\((w + 2)(w - 1)(w + 5)(w + 2)\\)
\\(\bigcirc\\) c. \\((w + 2)\\)
\\(\bigcirc\\) d. \\((w - 1)(w + 5)\\)

Explanation:

Step1: Factor the denominators

First, factor \( w^2 + w - 2 \). We need two numbers that multiply to \(-2\) and add to \(1\). Those numbers are \(2\) and \(-1\), so \( w^2 + w - 2=(w + 2)(w - 1) \).

Next, factor \( w^2 + 7w + 10 \). We need two numbers that multiply to \(10\) and add to \(7\). Those numbers are \(5\) and \(2\), so \( w^2 + 7w + 10=(w + 5)(w + 2) \).

Step2: Find the least common denominator (LCD)

The LCD of two fractions is the product of the highest power of each distinct factor that appears in the denominators. The factors of the first denominator are \((w + 2)\) and \((w - 1)\). The factors of the second denominator are \((w + 5)\) and \((w + 2)\). The distinct factors are \((w - 1)\), \((w + 2)\), and \((w + 5)\). The highest power of each factor: \((w - 1)^1\), \((w + 2)^1\), \((w + 5)^1\). So the LCD is \((w + 5)(w - 1)(w + 2)\).

Answer:

A. \((w + 5)(w - 1)(w + 2)\)