Problem 9 - Elementary and Intermediate Algebra

Question

Consider the following factorizations. $$\begin{array}{l}{18 x-36=2 \cdot 3 \cdot 3 \cdot(x-2)} \\\\{3 x-6=3(x-2)}\end{array}$$ a. What is the greatest number of times the factor 3 appears in any one factorization? b. What is the greatest number of times the factor \(x-2\) appears in any one factorization?

Step-by-Step Solution

Answer

3 appears twice; \(x-2\) appears once.

Step 1: Determine Factorization of First Expression

The first expression is \(18x - 36\). We factor it like so: \(18x - 36 = 2 \cdot 3 \cdot 3 \cdot (x - 2)\). Note here that the factor 3 appears twice, as it is expressed as \(3 \cdot 3\). The factor \(x-2\) appears once.

Step 2: Determine Factorization of Second Expression

The second expression is \(3x - 6\). Factorizing yields: \(3x - 6 = 3(x-2)\). In this expression, the factor 3 appears once, and \(x-2\) appears once.

Step 3: Analyze Factor Counts

Now compare the presence of each factor in the two given factorizations. For factor 3, the first expression has it appearing twice, while the second has it once. For factor \(x-2\), both expressions have it appearing once. Therefore, the greatest number of times factor 3 appears in any factorization is twice, while factor \(x-2\) appears at most once.

Key Concepts

Greatest Common Factor in Algebraic Expressions

The Greatest Common Factor (GCF) is an essential concept in algebra. It refers to the largest factor that is common to two or more numbers or terms. In algebra, we often encounter it while simplifying expressions.
When we factorize expressions like \(18x - 36\) and \(3x - 6\), identifying the GCF simplifies our work.

• For \(18x - 36\): The GCF is 18 because it is the largest number that divides both 18 and 36, which factors into \(2 \cdot 3 \cdot 3\).
• For \(3x - 6\): Here, the GCF is 3, the greatest factor that is common in both terms.

Identifying the GCF helps to simplify and solve polynomial equations easily by reducing them into simpler forms.

Understanding Polynomial Expressions

Polynomial expressions are a fundamental part of algebra. They consist of variables, coefficients, and exponents, and are expressed in terms like \(ax^n + bx^{n-1} + \ldots + cx + d\).
In our exercise, expressions \(18x - 36\) and \(3x - 6\) are polynomials. Let's break down the elements:

• Coefficients: In \(18x\), 18 is the coefficient; in \(3x\), 3 is the coefficient.
• Variable: \(x\) is the variable here, which can take different values.
• Constant term: In both expressions, 36 and 6 are constants.
  

Polynomial expressions like these often need factorization to simplify. Understanding their structure and how elements relate helps you manipulate them effectively.

Basics of Elementary Algebra in Factorization

Elementary Algebra is the foundational stone for understanding advanced mathematical concepts. It involves operations and manipulations with mathematical symbols. Factorization, a vital tool in algebra, is an operation where we express numbers or expressions as the product of their factors.
Consider two factorized forms:

• \(18x - 36 = 2 \cdot 3 \cdot 3 \cdot (x - 2)\)
• \(3x - 6 = 3 \cdot (x - 2)\)
  

Factors are the expressions that multiply to form the original expression. The operation involves knowing how to apply basic arithmetic operations — multiplication and division — and recognizing patterns in numbers and expressions. Mastering these skills builds the base of algebraic understanding and aids in solving more complex equations efficiently.