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The Greatest Common Factor (GCF) is the largest number and/or variable that divides each term in the polynomial without leaving a remainder. In the exercise, the given polynomial is \(27m^3 - 9m\). To find the GCF for the coefficients (27 and 9), consider the highest number that divides both, which is 9.

For the variables, you look at the common factors among them. Here, both terms have at least one \(m\). Therefore, the GCF for the variables is \(m\). Combining these, the overall GCF for \(27m^3 - 9m\) is \(9m\).

Understanding and finding the GCF correctly is crucial because it simplifies polynomial factoring. Once identified, you can factor it out to simplify the expression further.

The factored form of a polynomial is achieved by expressing it as a product of its factors. This is done by factoring out the Greatest Common Factor (GCF) from each term.

In the exercise, after identifying the GCF \(9m\), we factor it out from \(27m^3 - 9m\). This involves dividing each term by \(9m\) and then rewriting the polynomial. Here's how it's done:

• \(27m^3\) divided by \(9m\) results in \(3m^2\)
• \(9m\) divided by \(9m\) results in \(1\)

This leads us to the factored form \(9m(3m^2 - 1)\).

The factored form makes it easier to manipulate and solve polynomials, especially when factoring further or finding roots.

Polynomial factoring involves breaking down a polynomial into simpler 'factor' polynomials that, when multiplied together, give you the original polynomial. This skill is fundamental in algebra and is used to simplify expressions and solve polynomial equations.

The initial step in factoring is often finding the GCF and expressing the polynomial in its factored form. With the given polynomial \(27m^3 - 9m\), we identified the GCF \(9m\) and factored it, resulting in \(9m(3m^2 - 1)\).

Here are some key points to remember when factoring polynomials:

• Always look for the GCF first.
• After factoring out the GCF, check if the remaining polynomial can be factored further.
• The objective is to simplify the polynomial as much as possible.

Mastering polynomial factoring helps in many other algebraic concepts, such as solving quadratic equations and simplifying algebraic fractions.