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Factoring a polynomial often begins with finding the Greatest Common Factor (GCF).
The GCF is the largest factor that divides into all terms of the polynomial without leaving a remainder.
By identifying the GCF, you can simplify the polynomial by reducing each term in the expression.

When we look at the polynomial \(9x^3 - 9x\), we need to examine each term separately to determine the GCF.

• Consider the first term: \(9x^3\). The factors are: \(9\) and \(x^3\).
• Consider the second term: \(-9x\). The factors are: \(-9\) and \(x\).

When comparing the coefficients \(9\) and \(-9\), we see that the number \(9\) is a common factor.
For the variable terms, \(x\) is present in both terms, making it the greatest common variable factor.
Hence, the GCF of the entire expression is \(9x\).

By factoring out the GCF from the polynomial, we simplify it to: \(9x(x^2 - 1)\).
This makes it easier to perform further factorization steps.

The Difference of Squares is a useful formula in algebra, allowing us to factor expressions that have squares being subtracted.
The general formula states: if you have an expression of the form \(a^2 - b^2\), it can be factored into \((a + b)(a - b)\).

Let's apply this to the polynomial after we've factored out the GCF in the previous step.

• The expression \(x^2 - 1\) is a classic Difference of Squares scenario.
• Here, \(x^2\) is \(a^2\) and \(1\) can be considered as \(b^2\), because \(1 = 1^2\).

Using the Difference of Squares formula:

• Set \(a = x\).
• Set \(b = 1\).
• Then \(a^2 - b^2 = x^2 - 1 = (x + 1)(x - 1)\).

By recognizing and applying this formula, the expression \(x^2 - 1\) can be rewritten as \((x - 1)(x + 1)\).
This further simplifies the entire polynomial.

Complete factorization involves breaking down a polynomial into the product of its simplest factors.
In the case of the polynomial \(9x^3 - 9x\), we aimed for the most simplified, factorized form.

Initial steps involved finding the GCF and factoring out \(9x\), to yield \(9x(x^2 - 1)\).

Then, recognizing \(x^2 - 1\) as a Difference of Squares allowed us to rewrite it as \((x - 1)(x + 1)\).
Thus, the entirely factored form of the polynomial becomes:

• \(9x(x - 1)(x + 1)\)

This represents the complete factorization of the original polynomial \(9x^3 - 9x\).

It's important to check at each step that further factorization isn't possible, ensuring it is fully simplified.
By doing so, you'd confirm you achieved the polynomial's complete factorization that can't be broken down further.