Question

look at the expression below.
\\(\frac{2h + y}{9h^2 - y^2} - \frac{4h^2}{3h + y}\\)
which of the following is the least common denominator for the expression?
\\(\bigcirc\\) a. \\((3h + y)\\)
\\(\bigcirc\\) b. \\((3h - y)(3h + y)\\)
\\(\bigcirc\\) c. \\((3h + y)(3h + y)(3h - y)\\)
\\(\bigcirc\\) d. \\(3(h + y)(h - y)\\)

Explanation:

Step1: Factor the first denominator

The first denominator is \(9h^2 - y^2\). Using the difference of squares formula \(a^2 - b^2=(a - b)(a + b)\), where \(a = 3h\) and \(b = y\), we get \(9h^2 - y^2=(3h - y)(3h + y)\).

Step2: Identify the denominators

The denominators of the two fractions are \((3h - y)(3h + y)\) (from the first fraction) and \(3h + y\) (from the second fraction).

Step3: Determine the least common denominator (LCD)

The LCD is the product of the highest power of each distinct factor. The distinct factors are \((3h - y)\) and \((3h + y)\). The highest power of \((3h - y)\) is 1, and the highest power of \((3h + y)\) is 1. So the LCD is \((3h - y)(3h + y)\).

Answer:

B. \((3h - y)(3h + y)\)