Problem 93 - Algebra and Trigonometry

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

Step 1: Factorize the denominators.

Factorize the denominators to simplify the equations. The expression \(x^{2}-100\) can be factorized into \((x+10)(x-10)\), using the difference of squares formula (a² - b² = (a + b)(a - b)). The expression \(x^{2}-20 x+100\) can be factorized as \((x-10)^{2}\), using the square of a binomial formula (a² - 2ab + b² = (a - b)²).

Step 2: Find the common factors.

Determine the common factor from each factorized denominator. In this case, the common factor is \((x-10)\) since it appears in both denominators when factorized.

Step 3: Determine the least common denominator.

Finally, build the least common denominator, incorporating all the factors from both denominators, only duplicating the common factors once. Here, the factors are \((x+10)\), \((x-10)\), and \((x-10)\) again from the square. We only count \((x-10)\) once, so the least common denominator is \((x+10)(x-10)^{2}\).