Problem 49 - College Algebra and Calculus: An Applied Approach

Question

Find the least common denominator of the expressions.\(\frac{1}{x^{2}}, \frac{1}{x-1}, \frac{1}{x^{2}-x}\)

Step-by-Step Solution

Answer

The least common denominator of \(\frac{1}{x^{2}}\), \(\frac{1}{x-1}\), and \(\frac{1}{x^{2}-x}\) is \(x^{2}(x-1)\).

Step 1: Analyze the individual denominators

Let's look at each denominator. The first one is \(x^{2}\), the second one is \(x-1\), and the third one is \(x^{2}-x = x(x-1)\).

Step 2: Identify common factors

We notice that \(x\) is a common factor in the first and third denominators and \(x-1\) is a common factor in the second and third denominators.

Step 3: Determine the Least Common Denominator

The LCD is determined by taking the highest power of each factor that appears in the denominators. In our case, the highest power of \(x\) is 2 and for \(x-1\), the highest power is 1. Therefore, the LCD of \(\frac{1}{x^{2}}\), \(\frac{1}{x-1}\), and \(\frac{1}{x^{2}-x}\) is \(x^{2}(x-1)\).

Key Concepts

Rational Expressions

Rational expressions are similar to fractions, but they include polynomials in both the numerator and the denominator. Understanding rational expressions is crucial because they pop up frequently in algebra.
Just like regular fractions, these expressions need a common denominator when being added or subtracted. This commonality is referred to as the Least Common Denominator (LCD). It's essential to find the LCD to simplify the process of combining rational expressions.

• Example: In the given rational expressions \(\frac{1}{x^2}\), \(\frac{1}{x-1}\), and \(\frac{1}{x^2-x}\), the denominators are different.
• We need to identify a shared base to combine these expressions efficiently.

Recognizing rational expressions and practicing how to handle them will help you solve complex algebra problems comfortably.

Polynomial Factoring

Polynomial factoring is the process of breaking down a polynomial into simpler components called factors. These factors, when multiplied together, give back the original polynomial. Factoring is an important skill, especially when dealing with rational expressions, because it helps simplify expressions and find common denominators.

• To factor quadratics like \(x^2-x\), identify common factors in the terms. Here, both terms share the factor \(x\), so we factor it out: \(x(x-1)\).
• This reveals that \(x^2-x\) is composed of simpler linear expressions.

Factoring polynomials make it easier to identify LCD when combining expressions.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both, are algebraic expressions. They often involve polynomials and require special attention to manipulate and simplify, as they aren’t as straightforward as numerical fractions.
Working with algebraic fractions involves understanding how to simplify them by identifying common factors in denominators and numerators. Simplification often employs factoring and identifying the LCD, especially when adding or subtracting these fractions.

• For example, when dealing with algebraic fractions like \(\frac{1}{x^2}\), \(\frac{1}{x-1}\), and \(\frac{1}{x^2-x}\), finding the LCD involves factoring and comparing terms.
• Once the LCD, such as \(x^2(x-1)\), is determined, it becomes easier to combine and simplify the fractions.

Mastering algebraic fractions helps enhance algebraic fluency and assists in solving broader algebraic equations.