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The process of factoring begins with the identification of the greatest common factor, abbreviated as GCF. The GCF is the largest factor that divides two or more numbers or terms. To find the GCF, look at each term and determine the highest number that can divide them all without leaving a remainder.

When dealing with algebraic expressions, such as the exercise here \(25 b^{2} - 35\), we also look for variables that are common to each term; however, in this case, there are no common variables. So, the focus is solely on the coefficients. Both 25 and 35 have the number 5 as their GCF. This is the first and crucial step towards simplifying expressions, as it allows for a more streamlined representation of the equation.

Algebraic expressions are mathematical phrases that can contain numbers, variables (like \( b \)), and operation symbols. The expression \(25 b^{2} - 35\) is a simple example, where \( b \)) represents a variable, and the numbers 25 and 35 are coefficients attached to the terms of the expression.

Understanding how to manipulate these expressions is essential in algebra. The goal is to simplify the expression or rewrite it in a more useful form. This could involve processes like factoring, expanding, or solving for variables. When learning to factor, it is important to recognize common elements in each term of the expression, which is the essence of identifying the GCF.

Once the GCF is found, we use it to factor the polynomial. Factoring is the process of breaking down a complicated expression into simpler parts that, when multiplied together, give back the original expression. In our example, \(25 b^{2} - 35\), after finding that 5 is the GCF, we factor the polynomial by dividing each term by 5, which gives us \(5(5b^{2} - 7)\).

This could be visualized as separating a tray carrying two different types of fruits into two layers; with the bottom layer representing the common factor (like the GCF), and the top layer with compartments grouped by the quantity and type of fruit left after removing the common factor.

Polynomial factoring is not only a stepping stone to further algebraic manipulations but also a foundational skill for understanding more advanced math topics such as solving equations, simplifying fractions, and graphing functions.