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In algebra, coefficients are the numerical part of a term that contains a variable. For example, in the term \(20y^3\), the coefficient is 20. Coefficients help us understand the magnitude of the term they accompany. To solve problems like finding the greatest common factor (GCF), it's important to identify and work with coefficients independently from their variables. This enables us to break down complex terms into simpler components.
For the exercise involving the terms \(20 y^3, 28 y^2, 40 y\), first, separate the coefficients 20, 28, and 40 from the variable parts \(y^3, y^2, y\). This separation is key before proceeding with any further calculations.

Finding common factors is a crucial step when looking for the GCF. Common factors are the numbers that divide two or more numbers without leaving a remainder.
Let's use the coefficients from the earlier example: 20, 28, and 40. We need to list out the factors of each of these numbers.

• Factors of 20: 1, 2, 4, 5, 10, 20
• Factors of 28: 1, 2, 4, 7, 14, 28
• Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40

To find the common factors, we look for numbers that appear in all lists. In this case, the common factors are 1, 2, and 4. The greatest of these is 4, so the GCD of 20, 28, and 40 is 4.

Variable exponents refer to the power to which a variable is raised. In the terms \(20 y^3, 28 y^2, 40 y\), the exponents of the variable \(y\) are 3, 2, and 1, respectively. Identifying the variable part means recognizing that all terms share the same variable, \(y\), even if the exponents are different.
The key to finding the GCF involving variables is to use the lowest exponent of the common variable. Since \(y\) appears in each term, we look for the smallest power of \(y\) among \(y^3, y^2,\) and \(y\). This smallest exponent is 1. Therefore, the GCF of the variable part is \(y^1\) or simply \(y\).
Finally, we combine the GCD of the numerical coefficients (4) with the GCD of the variables (\