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When dealing with algebraic expressions, identifying the Greatest Common Factor (GCF) is the initial and crucial step in the process of factoring. The GCF is the largest number that can divide all the terms in an expression evenly. This means each term is divisible by this factor without leaving any remainder.
For example, in the expression \(3y^2 + 3y - 18\), we notice that each term can be divided by 3.

• The term \(3y^2\) divided by 3 gives \(y^2\).
• The term \(3y\) divided by 3 gives \(y\).
• The term \(-18\) divided by 3 gives \(-6\).

Once identified, the GCF, which is 3 in this exercise, can be "pulled out" of the expression. This leaves us with a simpler expression inside the parentheses: \(y^2 + y - 6\). Pulling out the GCF is like simplifying the expression, setting the stage for the next step of factoring the quadratic expression.

Quadratic expressions are polynomials that contain squared terms as their highest power. In algebra, these usually take the form of \(ax^2 + bx + c\). They can be quite common, and learning how to factor them is a key skill in algebra.

In the example we have, after removing the GCF, the quadratic expression simplifies to \(y^2 + y - 6\). The goal of factoring is to express the quadratic in terms of two binomials.

To factor \(y^2 + y - 6\), you need to find two numbers that:

• Add up to the middle term's coefficient, which is 1.
• Multiply to the last constant term, which is -6.

For this expression, the numbers 3 and -2 meet these criteria:

• 3 + (-2) = 1
• 3 × (-2) = -6

Thus, the expression \(y^2 + y - 6\) factors into \((y - 2)(y + 3)\). This decomposition into simpler expressions is what enables us to find the solution easily.

When an expression has been fully simplified into a product of its factors, it is said to be in its factored form. This form is helpful because it shows the expression as a multiplication of simpler terms, which can make solving equations more straightforward.

For our original expression \(3y^2 + 3y - 18\), the completely factored form retains the pulled-out GCF together with the factored quadratic expression, resulting in \(3(y - 2)(y + 3)\).

This final form shows us that the expression has been simplified to a most basic state, where no further factoring is possible. The factored form not only aids in simplifying expressions but also plays a vital role in solving equations and understanding roots of quadratic equations.