Problem 19 - Elementary and Intermediate Algebra

Question

Consider the following factorizations. $$\begin{aligned}&18 x-36=2 \cdot 3 \cdot 3 \cdot(x-2)\\\&3 x^{2}-3 x-6=3(x-2)(x+1) \end{aligned}$$ 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

a. 2 times b. 1 time

Step 1: Factorize the first expression

The expression given is \(18x - 36\). We can factor it fully as follows: \(18x - 36 = 2 \cdot 3 \cdot 3 \cdot (x - 2)\). Notice that 3 appears twice in this factorization (as \(3 \times 3\)) and \(x - 2\) appears once.

Step 2: Factorize the second expression

The expression provided is \(3x^2 - 3x - 6\). This can be written as \(3(x-2)(x+1)\). Here, 3 appears once and \(x - 2\) also appears once in this factorization.

Step 3: Count the occurrences of factor 3

Comparing both factorizations, 3 appears twice in \(18x - 36\) (as \(3 \times 3\)), and once in \(3x^2 - 3x - 6\). Hence, the greatest number of times the factor 3 appears in any one factorization is 2.

Step 4: Count the occurrences of factor (x - 2)

In each factorization, \(x - 2\) appears once. Therefore, the greatest number of times the factor \(x - 2\) appears in any one factorization is 1.

Key Concepts

Greatest Common Factor

Understanding the Greatest Common Factor (GCF) is a crucial part of algebraic factoring. The GCF is the largest factor that two or more expressions have in common. Identifying it helps simplify expressions, making them easier to work with. For example, in the expression \(18x - 36\), we can determine the GCF by listing out the factors of each term.

• The number 18 can be factored into \(2 \cdot 3 \cdot 3\).
• Meanwhile, the number 36 factors down to \(2 \cdot 3 \cdot 3 \cdot 2\).

Notice they share \(2 \cdot 3 \cdot 3\) as a factor, but the repeated factor that appears twice is 3 (since \(3\) is squared in both 18 and 36). The GCF for this expression is, therefore, 18.By factoring the expression as \(2 \cdot 3 \cdot 3 \cdot (x-2)\), we immediately apply the GCF to simplify and represent it in a more manageable form. Recognizing and using the GCF aids in breaking down complex algebraic problems into simpler parts.

Polynomial Factorization

Polynomial factorization breaks down a polynomial into simpler components, often revealed as a product of polynomials. This lets us solve equations more efficiently or simplify expressions.In multipart polynomial expressions, like \(3x^2 - 3x - 6\), each term is evaluated to find common factors. Initially, we notice that each term can be divided by 3, leading to the preliminary factorization into \(3(x^2 - x - 2)\).Next, the expression inside the parentheses, \(x^2 - x - 2\), is further factorized by analyzing factors of 2 that can sum or subtract to produce the middle term:

• The possible pairs are \((-1, 2)\) and \((1, -2)\).
• The pair \((-1, 2)\) leads to a middle term of -1 which needs adjustment.
• Trying \((x - 2)(x + 1)\) achieves the needed middle term and remains a factor of entire polynomial.

Therefore, the complete factorization of \(3x^2 - 3x - 6\) is \(3(x - 2)(x + 1)\). Recognizing such factor forms is central in algebra, allowing students to simplify expressions into a solvable format.

Algebraic Expressions

An algebraic expression is a mathematical phrase that combines numbers, variables, and operators (addition, subtraction, multiplication, and division). These expressions represent quantities and relationships in a mathematical form. They are structured and solved using fundamental algebraic principles.For instance, an expression like \(18x - 36\) is composed of a variable \(x\) and constant values 18 and 36. Operations within expressions use algebraic laws such as distribution and factorization to manipulate the expressions.

• The goal is often to simplify the expression to its simplest form.
• Simplified forms make it easier to insert values for variables, solve equations, or compare expressions.

Algebraic expressions allow us to represent real-world problems where relationships between numbers and unknowns exist. Mastery in handling them "unlocks" further challenging algebra concepts and mathematical logic applicable in numerous fields.