Problem 16 - Introductory Algebra for College Students

Question

Find the least common denominator of the rational expressions. $$\frac{7}{x^{2}-5 x-6} \text { and } \frac{x}{x^{2}-4 x-5}$$

Step-by-Step Solution

Answer

The least common denominator of the given fractions is \((x+1)(x-6)(x-5)\).

Step 1: Factor the Denominators

First, factor each of the quadratic expressions in the denominators. The factored form of \(x^{2}-5 x-6\) is \((x-6)(x+1)\). The factored form of \(x^{2}-4 x-5\) is \((x-5)(x+1)\).

Step 2: Identify Common and Uncommon Factors

Now identify the common and uncommon factors between the denominators. In this case, the common factor is \((x+1)\) and the uncommon factors are \((x-6)\) and \((x-5)\).

Step 3: Determine the Least Common Denominator

The least common denominator is the product of the common factor and all uncommon factors. So, the least common denominator for these fractions is \((x+1)(x-6)(x-5)\).