Problem 77 - 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 the denominators \(x^{2}-100\) and \(x^{2}-20 x+100\) is \((x^{2}-100)(x+10)\).

Step 1: Factoring the denominators

The first step in finding the least common denominator is to factor the given denominators. The expression \(x^{2}-100\) can be factored using the difference of squares rule, whereas the expression \(x^{2}-20 x+100\) can be factorized into binomial squares. The factorized forms are: \(x^{2}-100 = (x-10)(x+10)\) and \(x^{2}-20 x+100 = (x-10)^{2}\).

Step 2: Identifying the Least Common Denominator

Now that the denominators have been factored, the least common denominator can be identified. The LCD will be the expression that includes all the factors of both denominators. In this case, the factors are \(x-10\), \(x+10\), and \(x-10\). Therefore, the LCD is \((x-10)^{2}(x+10)\).

Step 3: Simplifying the Least Common Denominator

The final step is simplifying the LCD. The expression \((x-10)^{2}(x+10)\) simplifies to \((x^{2}- 100)(x+10)\) after expanding the binomial square.