Problem 93 - College Algebra

Question

Explain how to find the least common denominator for denominators of \(x^{2}-100\) and \(x^{2}-20 x+100\).

Step-by-Step Solution

Answer

The least common denominator for denominators of \(x^{2}-100\) and \(x^{2}-20x+100\) is \((x-10)^{2}(x+10)\).

Step 1: Factorize the Expressions

Begin by factorizing the denominators. The difference of two squares, \(x^{2}-100\), can be factored into \((x-10)(x+10)\). The perfect square trinomial, \(x^{2}-20x+100\), can be factored into \((x-10)^{2}\).

Step 2: Identify Common Factors

After factorizing the denominators, identify the highest power of each factor in both expressions. In this example, the factor \(x-10\) appears as \((x-10)\) in the first expression and \((x-10)^{2}\) in the second.

Step 3: Form the LCD

The LCD is the product of the highest power of all identified factors. So, the LCD for this problem would be \((x-10)^{2}(x+10)\), since the highest power of \(x-10\) found was 2, and \(x+10\) was also a factor in the expressions.