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The Greatest Common Factor, commonly abbreviated as GCF, is the largest factor shared by two or more numbers or terms. For students grappling with polynomial factorization, identifying the GCF is a critical first step, as it simplifies the expression, making subsequent factorization easier.

When facing a polynomial like the one in our exercise, the GCF is found by looking for the highest number and variable that evenly divides each term. In the given problem, both terms, 60 and 540, are divisible by 60, making it the GCF of this polynomial expression. By factoring it out, we reduce the polynomial to a simpler form inside the parentheses, setting the stage for further factorization techniques to be employed.

The 'difference of squares' is a special pattern in algebra where two perfect squares are subtracted from each other. It is one of the key concepts when working with polynomial expressions. This pattern has a unique property: it can always be factored into the product of two binomials that are conjugates of each other, that is, of the form \(a + b)\) and \(a - b)\).

In our textbook example, after factoring out the GCF, we observe that \(y^2 - 9\) fits the difference of squares pattern, where \(a = y\) and \(b = 3\), since both \(y^2\) and \(9\) are perfect squares and are being subtracted. Thus, \(y^2 - 9\) is factored into \(y + 3\) and \(y - 3\), revealing a foundational factoring skill that typically simplifies solving equations and inequalities.

Polynomial factorization is the process of expressing a polynomial as the product of its factors. These factors may include numbers, variables, and the powers of variables that, when multiplied together, give back the original polynomial. It's often a pivotal step in solving more complex algebraic expressions and equations.

The problem we addressed involves a two-step factorization process. First, the GCF is factored out, which simplifies the polynomial, then we identify and factor a difference of squares. Recognizing these patterns and understanding how to factor them can significantly aid in solving myriad polynomial-related problems. Mastery of these processes enables students to simplify expressions, find roots of equations, and understand deeper algebraic concepts such as graphing functions or analyzing algebraic structures.