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When factoring polynomials, the first step is to identify the Greatest Common Factor (GCF). The GCF of two or more polynomial terms is the largest expression that can evenly divide all the terms. Finding the GCF helps in simplifying the polynomial by reducing it to its smallest possible form.

To find the GCF, start by looking at the coefficients and the variables separately. Take the coefficients of each term. For instance, in the polynomial given, the coefficients are 12 and 24. The GCF of 12 and 24 is 12 because it's the largest number that can divide both 12 and 24 evenly.

Next, look at the variables. In our case, we have variables with powers: ^2 and ^3. The GCF for the variables is determined by the smallest power present in all terms. Here, the smallest power is ^2.

Combining these, the GCF of 12^2 and 24^3 is 12^2.

A polynomial is a mathematical expression consisting of variables, coefficients, and exponents that are combined using addition, subtraction, and multiplication. Each part of the polynomial separated by a '+' or '-' sign is called a term. For example, in the polynomial 12^2 + 24^3, there are two terms: 12^2 and 24^3.

Understanding each term is crucial because it allows us to see the overall structure of the polynomial. Each term consists of a coefficient (a number), a variable (such as ), and an exponent (the power to which the variable is raised).

• In the term 12^2, 12 is the coefficient, is the variable, and 2 is the exponent.
• In the term 24^3, 24 is the coefficient, is the variable, and 3 is the exponent.

Identifying these components helps in the process of factoring because it allows us to find common elements among the terms.

Factoring a polynomial completely means breaking it down into a product of simpler polynomials that cannot be factored further. It's like breaking down a complex object into its simplest building blocks.

Let's revisit the solution to the given exercise: we found the GCF of the polynomial 12^2 and 24^3, which is 12^2. Factoring out 12^2, we get:

Given that we factored out 12^2, we're left with:
12^2 (1 + 2n).

The expression inside the parentheses, 1 + 2n, is already in its simplest form and cannot be factored further. So, we've reached the completely factored form of the polynomial.

To summarize, we:

• Identified the GCF.
• Factored out the GCF from each term.
• Checked if the remaining expression could be factored further.

The final result is the product of the GCF and the simplest remaining expression.