Problem 7 - Elementary and Intermediate Algebra

Question

In the rational expression \(\frac{(x+2)(3 x-1)}{(x+2)(4 x+2)},\) the binomial \(x+2\) is a common _____ of the numerator and the denominator.

Step-by-Step Solution

Answer

Factor

Step 1: Identifying Common Factors

First, observe the rational expression \( \frac{(x+2)(3x-1)}{(x+2)(4x+2)} \). We need to identify the common factors in both the numerator and the denominator.

Step 2: Recognizing the Common Binomial

The term \((x+2)\) is present in both the numerator and the denominator as a separate factor in each.

Step 3: Determining the Term Type

Since \((x+2)\) is a factor in both the numerator and the denominator, it is considered a common factor of both parts of the rational expression.

Key Concepts

Common Factors

A common factor is an expression or number that appears in multiple terms within an equation or expression. In the context of rational expressions, we are often tasked with simplifying expressions by identifying common factors, as these allow us to reduce the expression to its simplest form. Take, for example, the rational expression \[ \frac{(x+2)(3x-1)}{(x+2)(4x+2)} \].When we look at both the numerator and denominator, we can observe that the binomial \( (x+2) \) appears in both places, making it a common factor.

• Finding common factors is key to simplifying rational expressions.
• Simplification makes the expressions easier to understand and use.
• This process of identifying and canceling common terms is fundamental in algebra.

Recognizing these common factors, particularly in polynomial expressions like the one we are dealing with, allows us to simplify the expression by removing or 'canceling' the common factors from both the numerator and the denominator. Just remember, a common factor can help reduce expressions to a more manageable and simple state.

Numerator and Denominator

The numerator and the denominator are crucial components of any fraction, including rational expressions. In any fraction, the numerator is the top number, while the denominator is the bottom number. In a rational expression like the one we are analyzing \[ \frac{(x+2)(3x-1)}{(x+2)(4x+2)} \],the entire expression can be considered as a fraction in which the numerator is made up of one polynomial and the denominator is made up of another.

• Numerator: This is the expression above the division line. In our case, \((x+2)(3x-1)\).
• Denominator: This is the expression below the division line, represented by \((x+2)(4x+2)\).

Understanding the roles of the numerator and denominator is fundamental in manipulating and simplifying rational expressions. Especially when both contain common factors, as simplifying such expressions often includes identifying these factors and canceling them out where possible. By simplifying, we make complex fractions much more straightforward and easier to interpret or solve.

Binomials

A binomial is a polynomial with exactly two terms. In algebra, you will often encounter binomials because they form the building blocks of many types of expressions. In our rational expression \[ \frac{(x+2)(3x-1)}{(x+2)(4x+2)} \],both the numerator and the denominator include the binomial \( (x+2) \). Recognizing binomials is essential because they frequently can be factored or used in simple arithmetic operations to simplify expressions.

• Binomials take the form \( a + b \) or \( a - b \).
• They are involved in operations like distribution or in forming quadratic equations.
• Common operations with binomials include expansion (distributing terms) and factoring.

In our example, the ability to spot the binomial \( (x+2) \) in both parts of the rational expression allows us to utilize it as a common factor. Understanding binomials can significantly aid in solving more complex algebraic problems, as they allow for streamlined calculation methods through processes like factoring and expansion. Knowing how to manipulate binomials effectively can make narrowing down solutions much more approachable.