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We are tasked with factoring out the greatest common factor (GCF) from the polynomial:

\[
30a^6b^6 - 28a^9b^9 + 28ab - 12a^9b
\]

### Step 1: Identify the GCF of the numerical coefficients
The coefficients are \(30, -28, 28, -12\). The greatest common factor of these numbers is 2.

### Step 2: Identify the GCF of the variables
- For \(a\): The powers of \(a\) are \(6, 9, 1, 9\). The smallest power is \(1\), so the GCF for \(a\) is \(a^1 = a\).
- For \(b\): The powers of \(b\) are \(6, 9, 1, 1\). The smallest power is \(b^1 = b\).

### Step 3: Factor the GCF out
The GCF of the polynomial is \(2ab\). Now, factor \(2ab\) out from each term:

\[
= 2ab(15a^5b^5 - 14a^8b^8 + 14 - 6a^8)
\]

### Final Answer:
The factored form of the polynomial is:

\[
2ab(15a^5b^5 - 14a^8b^8 + 14 - 6a^8)
\]

Let me know if you need further clarifications or details on any step!

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Here are 5 related questions for deeper understanding:
1. How do you find the greatest common factor of two or more terms?
2. Why is it important to factor out the GCF before attempting other factorization methods?
3. What happens when terms do not share any common factors?
4. How can factoring polynomials simplify solving polynomial equations?
5. Can factoring be used to simplify rational expressions?

**Tip:** Always start by factoring out the GCF from all terms—it simplifies the expression and helps with further factorization techniques!

Factor out the greatest common factor from the following polynomial: 30a^6b^6 - 28a^9b^9 + 28ab - 12a^9b

Learn how to factor out the greatest common factor (GCF) from the polynomial 30a^6b^6 - 28a^9b^9 + 28ab - 12a^9b. The solution involves identifying the GCF of numerical coefficients and variables. Suitable for students in Grades 9-11.

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