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Understanding how to factor polynomials is crucial when dealing with algebraic expressions, especially when simplifying or manipulating them. Factoring involves breaking down a complex polynomial into simpler parts, known as factors, that when multiplied together give back the original polynomial. This process is similar to finding what numbers can be multiplied together to create another number. For instance, the polynomial 2y^2-6y+9 can be factored to 2(y-3)^2, as shown in the example.

Correct factoring requires identifying common factors and applying reverse distribution, or as many learned it, 'undistributing'. This reverse process helps in identifying patterns such as the difference of squares, perfect square trinomials, or other recognizable factorable forms. Recognizing these patterns makes the task easier and more straightforward, aiding greatly in the simplification of algebraic fractions.

To simplify fractions, one must divide both the numerator (the top part of the fraction) and the denominator (the bottom part) by their greatest common factor. In the context of polynomial division, this means dividing each term of the numerator by the denominator, if possible. For algebraic expressions, this process can also involve canceling out common factors in both the numerator and the denominator.

Using the exercise example, after factoring 2(y-3)^2/y, the term y in the denominator cancels out one y from the numerator, leaving a simplified algebraic expression. Simplification makes the overall equation or expression more manageable and is an essential skill in solving algebra problems.

Algebraic expressions comprise numbers, variables, and arithmetic operations (addition, subtraction, multiplication, and sometimes division). Working with these expressions often involves performing a series of operations to simplify or manipulate them to solve equations or inequalities.

Division of algebraic expressions, as in the given exercise, requires understanding how these components interact. After factoring and simplifying, we can rewrite the expression in a simplified form. In this case, 2(y-3)^2/y gives us 2y - 6 + 9/y upon simplification. Each component of an algebraic expression must be treated with the mathematical operations that define the expression's overall behavior. Hence, understanding how to manipulate these expressions is critical in unlocking the broader application of algebra in various mathematical problems.