How to Factor by GCF?
In this step-by-step guide, you will learn more about factoring polynomials by the greatest common factor.
Factor by GCF: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Organize by degreeWrite terms from highest power to lowest power.
- Look for structureTry GCF, special products, grouping, or division depending on the expression.
- Check with featuresZeros, multiplicity, and end behavior should agree with your algebra.
Worked examples
Combine like terms
- Group x squared terms.
- Group x terms.
- Combine each group.
Factor a difference of squares
- Recognize a squared term minus a squared term.
- Use a^2 – b^2.
- Write conjugate factors.
Try one before moving on
Factor by GCF: pop-up practice
Related Topics
Step-by-step guide to factoring by \(GCF\)
The process of factoring polynomials includes expressing the polynomial as a product of its factors. Factoring polynomials helps to find the zeros of the polynomial expression or to find the values of the variables of the given expression.
There are several methods for factoring polynomials based on expressions. One of them is factoring based on the greatest common factor or \(GCF\). The \(GCF\) of a polynomial is the greatest monomial that divides the polynomial terms. Note that the \(GCF\) should be a factor of all terms in the polynomial.
The process of obtaining the greatest common factor for two or more terms involves two simple steps. First, divide each expression into its prime factors, and then take as many common factors as possible from the given expressions.
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Let us understand this using a simple expression of two terms \(15x^4 + 3x^3\). Here we split the terms into their prime factors \(15x^4 + 3x^3 = 5\times 3\times x\times x\times x\times x + 3\times x\times x\times x\). From these two terms, we can take the maximum common terms to get the greatest common factor. Here we have the maximum common factor as \(3x^3\), and hence \(15x^4 + 3x^3= 3x^3(5x + 1)\).
Factoring by \(GCF\) – Example 1:
Factor the polynomial. \(12y^3-27y^2\)
Solution: First, find the \(GCF\) of all of the terms:
\(12y^3=2×2×3×y×y×y\), \(27y^2=3×3×3×y×y\). The common factors are \(3×y×y\). And, the \(GCF\) is \(3y^2\).
So \(12y^3-27y^2 = 3y^2(4y-9)\).
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