Problem 52 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{3}-125}{x^{2}-25}$$

Step-by-Step Solution

Answer

The simplified form of the rational expression is \(\frac{x^{2}+5x+25}{x+5}\).

Step 1: Identify the patterns

Identify that both the numerator and the denominator have a difference of cubes and squares pattern. \[x^{3}-125\] is a difference of cubes and \[x^{2}-25\] is a difference of squares.

Step 2: Factor the numerator and the denominator

Factor the numerator using the difference of cubes formula \(a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})\) where \(a=x\) and \(b=5\). This results in \((x-5)(x^{2}+5x+25)\). Factor the denominator using the difference of squares formula \(c^{2}-d^{2}=(c+d)(c-d)\), where \(c=x\) and \(d=5\). Resulting to \((x+5)(x-5)\). The given expression becomes \[\frac{(x-5)(x^{2}+5x+25)}{(x+5)(x-5)}\].

Step 3: Cancel common factors

Notice that the common factor of \(x-5\) can be cancelled from both the numerator and the denominator, resulting in the final simplified expression of \[\frac{x^{2}+5x+25}{x+5}.\]