Question

the least common denominator of the rational numbers \\(\frac{x^2 - y^2}{(x^2 - 8x + 15)}\\) and \\(\frac{y^2}{15(x^2 - 6x + 9)}\\) is \
options: \
\\((x - 3)^3(x - 5)\\)\
\\(15(x - 3)(x - 5)\\)\
\\((x - 3)^2(x - 5)\\)\
\\(15(x - 3)^2(x - 5)\\)

Explanation:

Step1: Factor Denominators

First, factor the denominators. For the first denominator \(x^2 - 8x + 15\), we factor it as \((x - 3)(x - 5)\) (since \( - 3\times - 5=15\) and \(-3 + (-5)=-8\)). For the second denominator \(x^2 - 6x + 9\), we factor it as \((x - 3)^2\) (since it is a perfect square trinomial, \((x - 3)^2=x^2-6x + 9\)). The second fraction also has a constant factor of \(15\).

Step2: Find LCD of Polynomials and Constants

The least common denominator (LCD) of two rational expressions is found by taking the highest power of each factor that appears in the denominators. For the polynomial factors: the factor \((x - 3)\) appears with highest power \(2\) (from \((x - 3)^2\)) and the factor \((x - 5)\) appears with highest power \(1\) (from \((x - 3)(x - 5)\)). For the constant factor, the denominators have \(1\) (from the first fraction) and \(15\) (from the second fraction), so the constant part of the LCD is \(15\). Combining these, the LCD is \(15(x - 3)^2(x - 5)\).

Answer:

\(15(x - 3)^2(x - 5)\) (corresponding to the option "15(x - 3)^2(x - 5)")