what is the greatest common factor of 12a and 9a2

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11-03-2025

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Finding the greatest common factor (GCF) of algebraic expressions like 12a and 9a² involves identifying the largest number and the highest power of variables that divide both terms without leaving a remainder. Let's break down how to solve this.

Understanding Greatest Common Factor (GCF)

The greatest common factor, or GCF, is the largest number that divides exactly into two or more numbers. For example, the GCF of 12 and 18 is 6, because 6 is the largest number that divides evenly into both. We'll apply a similar concept, but with variables as well.

Finding the GCF of 12a and 9a²

Step 1: Find the GCF of the coefficients.

The coefficients are the numerical parts of the terms: 12 and 9. To find their GCF, we look for the largest number that divides evenly into both.

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 9: 1, 3, 9

The largest number common to both lists is 3. Therefore, the GCF of 12 and 9 is 3.

Step 2: Find the GCF of the variables.

The variables are 'a' and 'a²'. Remember that a² means a * a. The highest power of 'a' that divides into both 'a' and 'a²' is simply 'a'.

Step 3: Combine the GCFs

To get the overall GCF of 12a and 9a², we multiply the GCF of the coefficients and the GCF of the variables:

3 * a = 3a

Therefore, the greatest common factor of 12a and 9a² is 3a.

Illustrative Example

Let's demonstrate this with a simple division:

If we divide both 12a and 9a² by 3a, we get:

• 12a / 3a = 4
• 9a² / 3a = 3a

We're left with no remainders, confirming that 3a is indeed the greatest common factor.

Applying GCF in Algebra

Finding the greatest common factor is crucial in simplifying algebraic expressions, factoring polynomials, and solving equations. It allows us to express expressions in their simplest form, making further manipulations much easier.

Further Practice

Try finding the GCF of these pairs:

• 15x² and 25x
• 8b³ and 12b²
• 24xy and 36x²y²

Understanding how to find the GCF is a fundamental skill in algebra. Mastering this concept will lay a strong foundation for more advanced algebraic operations.