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Understanding the factors of the coefficients is crucial when finding the greatest common factor (GCF). In the given problem, we need to look at the coefficients 27, 45, and 9. A coefficient is a number that multiplies a variable. To find the factors, you break the number down into all possible divisors. For example:

- The factors of 27 are 1, 3, 9, and 27.
- The factors of 45 are 1, 3, 5, 9, 15, and 45.
- The factors of 9 are 1, 3, and 9.

After listing the factors, you identify the common ones across all coefficients. In this case, the common factors among 27, 45, and 9 are 1, 3, and 9. Out of these, 9 is the highest, so it is the greatest common factor for the coefficients.

Variables in algebra are symbols used to represent numbers, and they can have exponents. In our problem, the variable is y with different exponents: \(y^2\), \(y^3\), and \(y^4\). To find the GCF involving a variable, we need to look at the exponents. In algebra, the common variable factor is found by taking the variable with the lowest exponent.

Here,
\(y^2\) has an exponent of 2,
\(y^3\) has an exponent of 3,
and \(y^4\) has an exponent of 4.

The smallest exponent is 2, so the greatest common factor for the variables is \(y^2\).

Exponents are the numbers that indicate how many times a base is multiplied by itself. For example, in the term \(y^4\), 4 is the exponent and y is the base, meaning y is multiplied by itself 4 times. Understanding exponents is essential for simplifying expressions and finding the GCF.

When dealing with exponents in algebraic expressions, the GCF involves taking the smallest exponent present when combining like terms. In our example with terms \(27 y^2\), \(45 y^3\), and \(9 y^4\), the common variable is y, and the smallest exponent is 2. Therefore, the term representing the GCF for the variable part will be \(y^2\).

Combining the GCF of the coefficients (9) and the smallest exponent variable (\(y^2\)), gives us the final greatest common factor: \(9 y^2\). This approach simplifies problem-solving and aids in understanding algebra more deeply.