GCF Factoring Calculator
Factor out the greatest common factor (GCF) from polynomials. The essential first step in all factoring problems.
Enter Polynomial Terms
Enter coefficient and power for each term (e.g., 6x³ → coefficient: 6, power: 3)
GCF Factoring Process:
- Find GCF of all coefficients
- Find lowest power of each variable
- Combine to form the complete GCF
- Divide each term by the GCF
Understanding GCF Factoring
Factoring out the Greatest Common Factor (GCF) is the foundation of all polynomial factoring. It should always be your first step before attempting any other factoring technique.
What is a GCF?
The GCF is the largest expression that divides evenly into all terms of a polynomial. It includes:
- The greatest common factor of all numerical coefficients
- The lowest power of each variable that appears in all terms
Step-by-Step Process:
Example: Factor 6x³ + 9x²
Step 1: GCF of coefficients (6, 9) = 3
Step 2: Lowest power of x is x²
Step 3: Complete GCF = 3x²
Step 4: Divide each term by 3x²
• 6x³ ÷ 3x² = 2x
• 9x² ÷ 3x² = 3
Answer: 3x²(2x + 3)
Why Factor Out GCF First?
- Simplifies the polynomial, making other factoring easier
- Reduces the size of numbers you're working with
- May reveal additional factoring opportunities
- Required for complete factorization
- Makes calculations less error-prone
Common Mistakes to Avoid:
- Forgetting to factor out the GCF before other methods
- Not finding the complete GCF (missing variables or powers)
- Factoring out too much or too little
- Making division errors when factoring out the GCF
- Not checking if the remaining polynomial can be factored further
Frequently Asked Questions
What does GCF stand for?
GCF stands for Greatest Common Factor. It's the largest factor that divides evenly into all terms of an expression. In polynomials, it includes both numerical coefficients and variable parts.
Do I always need to factor out the GCF?
Yes! Factoring out the GCF should always be your first step in any factoring problem. Even if the GCF is 1 (meaning there's no common factor), checking for it is important. It simplifies subsequent factoring steps.
How do I find the GCF of coefficients?
List the factors of each coefficient and find the largest factor that appears in all lists. Alternatively, use the Euclidean algorithm (repeated division) to find the GCF of two numbers, then repeat for additional numbers.
What if the terms have different variables?
Only include variables in the GCF if they appear in ALL terms. For example, in 6x²y + 9x, the GCF is 3x (not 3x²y) because y doesn't appear in both terms.
Can the GCF include negative numbers?
Technically yes, but it's conventional to factor out a positive GCF. However, if all terms are negative or if it simplifies the expression, you might factor out a negative GCF.
What if there's no common factor?
If the only common factor is 1, then there's no GCF to factor out. This is fine - just proceed with other factoring methods. Not all polynomials have a GCF greater than 1.
How do I check if I factored correctly?
Multiply the GCF by each term inside the parentheses. If you get back the original polynomial, your factoring is correct. This distributive property check is essential for verification.
Should I factor the expression in parentheses further?
Yes! After factoring out the GCF, always check if the remaining expression in parentheses can be factored further using other methods (difference of squares, trinomial factoring, etc.). Complete factorization includes all possible factoring steps.