Question

find the lcd for the following rational expressions.\\(
\frac{6y - 1}{y^2 - 9}, \frac{1 - 2y}{y^2 + 6y + 9}\\)

lcd = \\(\square\\) (simplify your answer.)

Explanation:

Step1: Factor the denominators

First, factor \(y^2 - 9\). Using the difference of squares formula \(a^2 - b^2=(a + b)(a - b)\), we have \(y^2 - 9=(y + 3)(y - 3)\).
Next, factor \(y^2 + 6y + 9\). Using the perfect square trinomial formula \(a^2+2ab + b^2=(a + b)^2\), we get \(y^2 + 6y + 9=(y + 3)^2\).

Step2: Determine the LCD

The LCD of two rational expressions is the least common multiple (LCM) of their denominators. To find the LCM of \((y + 3)(y - 3)\) and \((y + 3)^2\), we take the highest power of each factor that appears.
For the factor \((y + 3)\), the highest power is \((y + 3)^2\). For the factor \((y - 3)\), the highest power is \((y - 3)^1\).
So, the LCD is \((y + 3)^2(y - 3)\).

Answer:

\((y + 3)^2(y - 3)\)