Problem 38 - Elementary and Intermediate Algebra

Question

Find the GCF of each list of terms. $$ 2(y-1), 5(y-1) $$

Step-by-Step Solution

Answer

The GCF is \((y-1)\).

Step 1: Identify the Common Factor

Begin by looking at the expressions \(2(y-1)\) and \(5(y-1)\). Notice that both terms include a common factor, \((y-1)\).

Step 2: Determine the Numerical Greatest Common Factor

The remaining coefficients outside of the common factor \((y-1)\) are 2 and 5. The greatest common factor (GCF) of 2 and 5 is 1 since 2 and 5 are prime numbers.

Step 3: Multiply the Common Factor by Numerical GCF

Since the numerical GCF is 1, the greatest common factor of the terms is simply the common factor \((y-1)\) itself.

Key Concepts

Common Factor

A common factor is a value or an expression that is present in every term of a list or an equation. It is the part that you can "factor out" from an expression. For example, in mathematics, when you look at terms involving variables and constants, you often try to identify common factors to simplify the expression. In the problem, we observe that both terms, \(2(y-1)\) and \(5(y-1)\), share a common factor, \((y-1)\). This means that \((y-1)\) is a piece of each term that can be divided out, helping to identify the greatest common factor more easily. Recognizing common factors is crucial because it allows us to simplify expressions and solve equations more efficiently. Identifying the commonality among terms reduces the complexity of mathematical problems.

Numerical Coefficients

Numerical coefficients are the numerical parts of a term in an expression. They multiply the variables or factors attached to them. In algebra, a key part of finding the greatest common factor involves examining these numerical coefficients. When working with expressions like \(2(y-1)\) and \(5(y-1)\), the numbers 2 and 5 are the coefficients. These are what precede the common factor in each term. To find the greatest common factor, or GCF, of the entire expression, you first need to determine the GCF of these numerical coefficients. In this example, 2 and 5 are considered, but they do not have any common factors other than 1. Thus 1 is their GCF. Understanding numerical coefficients helps in simplifying expressions and solving for the highest dividing term shared among coefficients.

Prime Numbers

Prime numbers are whole numbers greater than 1 that have no divisors other than 1 and themselves. They play a significant role in understanding and finding the greatest common factor of numerical coefficients. In this exercise, the coefficients 2 and 5 are both prime numbers. Since prime numbers are only divisible by 1 and themselves, the only common factor they share is 1. Identifying that 2 and 5 are primes can speed up the process of determining their greatest common factor, because once we see they're primes, we know immediately that their only shared factor is 1. Understanding prime numbers helps in simplifying calculations since they can quickly determine divisibility and commonality in coefficient analysis.