Problem 7 - Elementary Algebra

Question

For exercises 1-12, use prime factorization to find the least common denominator. $$ \frac{2}{x^{2}-7 x} ; \frac{3}{3 x-21} $$

Step-by-Step Solution

Answer

The least common denominator is \(3x(x-7)\).

Step 1: Factor the Denominators

First, factor each denominator completely.For the first fraction, the denominator is \(x^{2}-7x\). Factor out the greatest common factor (GCF), which is \(x\): \(x^{2}-7x = x(x-7)\).For the second fraction, the denominator is \(3x-21\). Factor out the GCF, which is 3: \(3x-21 = 3(x-7)\).

Step 2: List the Prime Factors

Write out the prime factors of each denominator:First denominator \(x(x-7)\) has prime factors: \(x\) and \(x-7\).Second denominator \(3(x-7)\) has prime factors: \(3\) and \(x-7\).

Step 3: Identify the Least Common Denominator (LCD)

To find the LCD, include each factor the greatest number of times it occurs in any one factorization. The factors are \(3\), \(x\), and \(x-7\). The highest power of each factor in any denominator is: - \(3\) (appears once)- \(x\) (appears once)- \(x-7\) (appears once)Thus, the LCD is: \(3 \times x \times (x-7) = 3x(x-7)\).

Key Concepts

Prime Factorization and Least Common Denominator (LCD)

Understanding how to find the least common denominator (LCD) is crucial for working with fractions and rational expressions. The process starts with **prime factorization**, which is breaking down each term into its prime factors. You often need to find the **greatest common factor** (GCF) first, as it helps you factor polynomials and other expressions.

In the given exercise, both denominators were factored using prime factorization techniques:

• The first denominator was broken down into: \(x^{2} - 7x = x(x-7)\).
• The second denominator was broken down into: \(3x - 21 = 3(x-7)\).

Next, we listed the prime factors:

• The first fraction had: \(x\) and \(x-7\).
• The second fraction had: \(3\) and \(x-7\).

The goal is to identify the least common denominator by including each factor the greatest number of times it occurs in any one factorization. For this exercise, the LCD turned out to be \(3x(x-7)\).

Factoring Polynomials

Factoring polynomials is a vital skill that simplifies expressions and helps in various algebra topics including finding the LCD. Factoring involves rewriting a polynomial as a product of simpler polynomials. This often starts with identifying and factoring out the greatest common factor (GCF).

In the exercise's first step, we factored the polynomial denominators:

• For \(x^{2} - 7x\), the GCF is \(x\), resulting in \(x(x-7)\).
• For \(3x - 21\), the GCF is 3, resulting in \(3(x-7)\).

The GCF helps simplify the expressions into manageable parts, which is helpful when identifying the LCD or simplifying fractions. Always look for common factors first before moving on to other factoring techniques like grouping or using special polynomials.

Greatest Common Factor (GCF)

The concept of the greatest common factor (GCF) is fundamental in simplifying algebraic expressions and finding the least common denominator. The GCF is the largest factor shared by two or more numbers or terms.

When we look at the exercise, we first determined:

• The GCF of \(x^{2} - 7x\) was \(x\).
• The GCF of \(3x - 21\) was 3.

Factoring out these GCFs made it easier to identify the terms’ prime factors, ultimately helping us find the LCD. By extracting the GCF in the initial step of the factorization, you simplify the remaining expression and make finding common terms easier.
Understanding GCF is not just about simplifying; it connects deeply with other concepts like LCD and polynomial factorization, providing a comprehensive toolset for tackling various algebraic problems.