Problem 15 - Introductory Algebra for College Students

Question

Find the least common denominator of the rational expressions. $$\frac{3}{x^{2}-x-20} \text { and } \frac{x}{2 x^{2}+7 x-4}$$

Step-by-Step Solution

Answer

The least common denominator of the given rational expressions is \((x+4)*(x-5)*(2x-1)\)

Step 1: Factorise the denominators

The first step is to factorise the denominators. The factors of \(x^{2}-x-20\) are \((x+4)\) and \((x-5)\), because \(-5 * 4 = -20\) and \(-5 + 4 = -1\). Similarly, the factors of \(2x^2+7x-4\) are \((2x-1)\) and \((x+4)\), because \(1*4* 2 = -8\) and \(-8 + 7 = -1\).

Step 2: Identify the common factors

Having factored the denominators, we can now identify the common factors. From the factorised denominators, it is clear that \((x+4)\) is the common factor.

Step 3: Find the least common denominator

The least common denominator is obtained by multiplying the unique factors from both expressions. Thus, the least common denominator for the given rational expressions is \((x+4)*(x-5)*(2x-1)\).