Problem 4 - Elementary and Intermediate Algebra

Question

Fill in the blanks. The terms \(x(x-1)\) and \(4(x-1)\) have the common __________ factor \(x-1\)

Step-by-Step Solution

Answer

Common factor is \(x-1\).

Step 1: Understanding the Problem

We need to find the common factor between the terms \(x(x-1)\) and \(4(x-1)\). A common factor means a factor that is present in both terms.

Step 2: Breaking Down the Terms

Let's look at each term individually. The first term is \(x(x-1)\), which includes the factor \(x-1\). The second term is \(4(x-1)\), which also includes the factor \(x-1\).

Step 3: Identifying the Common Factor

We notice that both terms \(x(x-1)\) and \(4(x-1)\) have the factor \(x-1\) in common. This makes \(x-1\) the common factor between the two expressions.

Key Concepts

Common factor

A common factor is an element that is shared by two or more terms in an algebraic expression. To better understand what a common factor is, let's break down the expressions. Consider the terms \(x(x-1)\) and \(4(x-1)\). In these expressions, the factor \(x-1\) appears in both terms. This means that \(x-1\) is a common factor.

• The common factor is crucial for simplifying expressions.
• It helps in factoring expressions further by identifying shared elements.
• Having a common factor makes it possible to simplify complex expressions into simpler ones.

Recognizing common factors can simplify operations in algebra and make problem-solving much easier.

Algebraic expressions

An algebraic expression is a combination of numbers, variables, and operations. Algebraic expressions can represent real-world quantities in a mathematical form. For example, \(x(x-1)\) and \(4(x-1)\) are algebraic expressions. They contain variables \(x\), constants, and operations like multiplication and subtraction.Some characteristics of algebraic expressions include:

• They can represent both simple and complex mathematical relationships.
• They can be evaluated by substituting values for the variables.
• Expressions can be manipulated using mathematical operations such as addition, subtraction, multiplication, and division.

Learning to work with algebraic expressions is fundamental in developing problem-solving skills, as they model real-world situations in a clear, logical way.

Polynomial factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. Factoring polynomials is crucial because it simplifies expressions and helps solve equations. Consider the polynomial terms \(x(x-1)\) and \(4(x-1)\) from the given problem. They can be factored to reveal the common factor \(x-1\). Key points about polynomial factorization:

• It reduces complex polynomials into simpler ones, making them easier to work with.
• Helps in solving polynomial equations by finding roots.
• It sheds light on the structure of polynomials, showing how they can be broken down into their component parts.

Understanding polynomial factorization equips students with the ability to simplify and solve not only polynomials but also many other types of algebraic expressions.