How Do You Factor Out Factors?

Factoring out factors involves identifying a common factor shared by all terms in an expression and then rewriting the expression as a product of that common factor and a new expression inside parentheses.

Steps to Factor Out Factors:

1. Identify the Greatest Common Factor (GCF): Determine the largest number or expression that divides evenly into all terms of the expression you are trying to factor. This involves finding the greatest common factor for the coefficients (numerical part) and the lowest power of any common variables.

2. Divide Each Term by the GCF: Divide each term in the original expression by the GCF that you identified in step 1.

3. Rewrite the Expression: Write the GCF outside of a set of parentheses. Inside the parentheses, write the results of dividing each term by the GCF (from step 2). This represents the original expression in factored form.

4. Check Your Work (Optional): Distribute the GCF back into the parentheses. The result should be the original expression. This step is a way to verify that you have correctly factored out the common factor.

Examples:

Example 1: Factoring out a Numerical Factor

Factor the expression: 6x + 12

1. GCF: The greatest common factor of 6 and 12 is 6.

2. Divide: Divide each term by 6:

   • 6x / 6 = x
   • 12 / 6 = 2

3. Rewrite: Write the GCF (6) outside the parentheses, and the results of the division (x and 2) inside:

   6(x + 2)

4. Check: 6(x + 2) = 6x + 12 (Correct)

Example 2: Factoring out a Variable Factor

Factor the expression: x² + 5x

1. GCF: The greatest common factor of x² and 5x is x.

2. Divide: Divide each term by x:

   • x² / x = x
   • 5x / x = 5

3. Rewrite: Write the GCF (x) outside the parentheses, and the results of the division (x and 5) inside:

   x(x + 5)

4. Check: x(x + 5) = x² + 5x (Correct)

Example 3: Factoring out a Combination of Numerical and Variable Factors

Factor the expression: 4x³ + 8x² - 12x

1. GCF: The greatest common factor of 4, 8, and 12 is 4. The greatest common factor of x³, x², and x is x. Therefore, the overall GCF is 4x.

2. Divide: Divide each term by 4x:

   • 4x³ / 4x = x²
   • 8x² / 4x = 2x
   • -12x / 4x = -3

3. Rewrite: Write the GCF (4x) outside the parentheses, and the results of the division (x², 2x, and -3) inside:

   4x(x² + 2x - 3)

4. Check: 4x(x² + 2x - 3) = 4x³ + 8x² - 12x (Correct)

Key Considerations:

• Always look for the greatest common factor to fully factor the expression in one step.
• Be mindful of signs (positive and negative) when dividing.
• If there are no common factors (other than 1), the expression cannot be factored using this method.