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show me the variable exppression and give me the gcf

Let's analyze each of the three expressions to find the **Greatest Common Factor (GCF)**.

### 1. Expression 30: 
\[
-24x^6 - 4x^4 + 12x^3 + 8x^2
\]
The coefficients are: -24, -4, 12, 8.
- The GCF of these coefficients is **4** (since 4 is the largest number that divides all of them).

For the variable \( x \):
- The lowest power of \( x \) is \( x^2 \), so the GCF for the variable part is \( x^2 \).

Thus, the **GCF** for this expression is:
\[
4x^2
\]

### 2. Expression 32: 
\[
4a^2b^3 + 4a^2b - 7a - 8
\]
We analyze the coefficients: 4, 4, -7, -8.
- The GCF of the coefficients is **1** (since no number larger than 1 divides all terms).

For the variables:
- The first two terms both have \( a^2 \), while the last two terms only have \( a \), and the variables \( b^3 \) and \( b \) are only in the first two terms. Since the last two terms do not contain \( b \), there is no common variable factor across all terms.

Thus, the **GCF** for this expression is:
\[
1
\]

### 3. Expression 34: 
\[
-x^3y^6 - 10x^4y - 9y^2 + 9
\]
The coefficients are: -1, -10, -9, 9.
- The GCF of these coefficients is **1** (since no number larger than 1 divides all terms).

For the variables:
- Not all terms have the variables \( x \) or \( y \), and the last term (constant 9) has no variables. Hence, there is no common variable factor.

Thus, the **GCF** for this expression is:
\[
1
\]

---

### Summary of GCFs:
- For expression 30: **4x²**
- For expression 32: **1**
- For expression 34: **1**

Would you like more details or explanations on any part of this? 

### Here are 5 related questions:
1. What is the purpose of finding the GCF in expressions?
2. How do you factor an expression using the GCF?
3. What is the GCF of more complex expressions involving different variables?
4. How does the degree of the variables affect the GCF in polynomial expressions?
5. Can you explain how the GCF is related to factoring quadratic or cubic equations?

### Tip:
When factoring polynomials, always check for the GCF first before attempting more complex factorization methods!

show me the variable expression and give me the gcf

This page covers how to find the Greatest Common Factor (GCF) in polynomials such as -24x^6 - 4x^4 + 12x^3 + 8x^2 and others. Key mathematical concepts involve factoring, identifying common variables, and applying the GCF theorem. Suitable for Grades 9-12.

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