How can a greatest common factor be separated from an expression?

To separate a greatest common factor (GCF) from an expression, you must identify the GCF of all the terms, and then factor it out. This involves dividing each term by the GCF and expressing the result in factored form.

Here's a step-by-step process, incorporating information from the "Factoring A GCF From an Expression Lessons" reference:

Factoring Out a Greatest Common Factor (GCF)

1. Identify the GCF: First, determine the greatest common factor that divides into all terms of the expression. This involves looking at the coefficients (numbers) and variables in each term.

   • For example, if you have the expression 3x³ + 27x² + 9x, the GCF of the coefficients (3, 27, and 9) is 3. Also, each term contains 'x', so 'x' is also part of the GCF. The smallest exponent is 1, so the GCF is 3x.

2. Divide Each Term by the GCF: Next, divide each term in the original expression by the GCF you have identified.

   • Continuing with the example:
     • 3x³ / 3x = x²
     • 27x² / 3x = 9x
     • 9x / 3x = 3

3. Rewrite in Factored Form: Now, write the GCF outside of parenthesis and the results of the division inside the parenthesis.

   • In our example: 3x (x² + 9x + 3)

Summary Table

Steps	Description	Example (using 3x³ + 27x² + 9x)
1. Find the GCF	Identify the largest factor that divides all terms evenly.	The GCF of 3x³, 27x², and 9x is 3x.
2. Divide by GCF	Divide each term of the expression by the GCF.	3x³ / 3x = x², 27x² / 3x = 9x, 9x / 3x = 3
3. Write in Factored Form	Express the original expression as the GCF multiplied by the results of division, enclosed in parentheses.	3x (x² + 9x + 3)

By following these steps, you successfully separate the greatest common factor from an expression, representing it in its factored form.