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When working with polynomials, one of the first steps in simplifying or factoring them is finding the Greatest Common Factor (GCF). The GCF is the largest factor that divides each of the terms in the polynomial without leaving a remainder.

To identify the GCF:

• Analyze each term in the polynomial individually.
• Identify the factors in common among these terms.

In the exercise, each term includes the factor \(y-2\), making it the GCF for the expression \(4(y-2)^2 + 3(y-2)\).

Finding the GCF first is crucial as it simplifies further steps in the polynomial factorization process. By factoring out the GCF, you can simplify the expression and make it easier to work with.

Factoring polynomials involves writing the polynomial as a product of its factors. It is an essential technique in algebra and reveals the roots of the polynomial when set to zero.

Steps to factor a polynomial include:

• Identify and factor out the GCF from all terms.
• Express the remaining simplified terms as a polynomial inside a set of parentheses.

For the expression in our exercise, once we factor out the GCF \(y-2\), we are left with \(4(y-2) + 3\) inside the parentheses. This transforms the original complex expression into a simpler factored form: \( (y-2)(4(y-2) + 3) \).

Factoring polynomials not only simplifies them but makes it easier to solve equations, especially when dealing with higher-degree polynomials. It is also a valuable tool in simplifying expressions and computing integrals in calculus.

An algebraic expression is a combination of numbers, variables, and operations. These expressions are the building blocks of algebra and help to represent real-world problems mathematically.

In the expression \(4(y-2)^2 + 3(y-2)\):

• Variables are represented ( extit{like }\(y\)).
• Expressions inside parentheses ( extit{like }\((y-2)\)) are treated as single units.
• Operations include multiplication and addition.

By recognizing these structure elements, simplifying and solving algebraic expressions becomes straightforward.

In the factoring exercise, understanding the form and elements of the algebraic expression helps to correctly identify and pull out common factors, leading to a simplified and factored final result. Understanding how to manipulate these expressions is key in solving algebraic equations and is foundational for more advanced mathematical concepts.