How Do You Take Out the Common Factor?

To take out (or factor out) the common factor from an expression, you essentially reverse the distributive property. Here's how:

1. Identify the Greatest Common Factor (GCF):

• Look for the largest number that divides evenly into all the numerical coefficients in the expression.
• Identify the variable(s) that are common to all terms. If a variable is present in all terms, take the lowest power of that variable.

2. Divide Each Term by the GCF:

• Divide each term in the original expression by the GCF you identified.

3. Write the Factored Expression:

• Write the GCF outside a set of parentheses.
• Inside the parentheses, write the result of dividing each term by the GCF.

Example 1: Factoring out a numerical GCF

Let's factor the expression: 12x + 18y

• GCF: The greatest common factor of 12 and 18 is 6.
• Divide:
  • 12x / 6 = 2x
  • 18y / 6 = 3y
• Factored Expression: 6(2x + 3y)

Example 2: Factoring out a variable GCF

Let's factor the expression: x³ + x²

• GCF: The greatest common factor is x² (the lowest power of x present in both terms).
• Divide:
  • x³ / x² = x
  • x² / x² = 1
• Factored Expression: x²(x + 1)

Example 3: Factoring out a numerical and variable GCF

Let's factor the expression: 8a²b - 12ab²

• GCF: The greatest common factor is 4ab. (4 is the GCF of 8 and 12; 'a' is common with power 1; 'b' is common with power 1)
• Divide:
  • 8a²b / 4ab = 2a
  • -12ab² / 4ab = -3b
• Factored Expression: 4ab(2a - 3b)

In Summary:

Factoring out the common factor involves finding the greatest common factor among all terms in an expression, dividing each term by that factor, and then rewriting the expression as the product of the GCF and the resulting expression in parentheses. This simplifies expressions and is a foundational skill in algebra.