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How to Factor GCF Step by Step?

Published in Factoring Algebra 2 mins read

To factor the Greatest Common Factor (GCF) step-by-step, follow these procedures:

Steps to Factor GCF

Here's how you can factor the GCF, explained clearly and with examples:

  1. Prime Factorization: Break down each coefficient into its prime factors and expand the variables. This means writing out all the variables with their respective exponents as a product of variables.

    • Example: For 12x²y it becomes 2 × 2 × 3 × x × x × y and for 18xy² it becomes 2 × 3 × 3 × x × y × y

  2. Identify Common Factors: List all the factors of each term, aligning common factors in a column. This will make it easy to see which factors are shared among all terms.

    • Example:

      Term Prime Factors
      12x²y 2 × 2 × 3 × x × x × y
      18xy² 2 × 3 × 3 × x × y × y

  3. Bring Down Common Factors: Identify and bring down factors that are common to all expressions.

    • Example: Looking at the table above, 2, 3, x and y are present in both terms. So they are brought down.

  4. Multiply Common Factors: Finally, multiply the common factors you brought down in the previous step. This product is the GCF.

    • Example: From our example, the common factors are 2 × 3 × x × y, which equals 6xy. Therefore the GCF of 12x²y and 18xy² is 6xy.

Example Walkthrough

Let's apply these steps to factor the GCF of 24a³b² + 36a²b³

  1. Prime Factorization:

    • 24a³b² = 2 × 2 × 2 × 3 × a × a × a × b × b
    • 36a²b³ = 2 × 2 × 3 × 3 × a × a × b × b × b

  2. List Common Factors:

    Term Prime Factors
    24a³b² 2 × 2 × 2 × 3 × a × a × a × b × b
    36a²b³ 2 × 2 × 3 × 3 × a × a × b × b × b

  3. Bring Down Common Factors: Common factors are 2, 2, 3, a, a, b, and b.

  4. Multiply Common Factors: 2 × 2 × 3 × a × a × b × b = 12a²b². Thus, the GCF is 12a²b².

Practical Tips

  • Be thorough when finding prime factors.
  • Double-check to make sure all common factors are identified.
  • Pay careful attention to variable exponents.

By carefully following these steps, you can effectively find and factor out the GCF of any set of terms.

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