GCF as First Step in Factoring Strategy
Factoring out the greatest common factor (GCF) is always the first step in any polynomial factoring strategy, as taught in Grade 11 enVision Algebra 1 (Chapter 7: Polynomials and Factoring). Before attempting trinomial factoring, difference of squares, or other patterns, students check whether a GCF exists across all terms. Removing the GCF simplifies the remaining polynomial, often revealing a recognizable pattern and ensuring the final answer is fully factored. Skipping this step is the most common cause of incomplete factoring.
Key Concepts
When factoring a polynomial, always ask first, "Is there a greatest common factor?" If there is, factor it first. This is the initial step in any factoring strategy. Factoring out the GCF simplifies the remaining polynomial and ensures the expression is in its most factored form.
Common Questions
Why is factoring out the GCF always the first step?
Factoring the GCF first simplifies the remaining polynomial and ensures the expression reaches its most completely factored form. Skipping it often leads to incomplete answers.
What is the greatest common factor of a polynomial?
The GCF of a polynomial is the largest monomial that divides evenly into every term of the polynomial — it includes the largest shared numerical factor and any shared variable factors.
How do you factor out the GCF from a polynomial?
Identify the GCF of all terms, write it outside parentheses, then divide each term by the GCF to get the expression inside the parentheses.
What happens if you do not factor out the GCF first?
The remaining polynomial will still have a common factor, meaning it cannot be factored completely using other techniques alone.
Can the GCF be a variable expression?
Yes. If every term contains at least x², then x² is part of the GCF and should be factored out.
What should you do after factoring out the GCF?
Check whether the resulting factor inside the parentheses can be factored further using difference of squares, trinomial factoring, or other patterns.