5.1 Factoring the Greatest Common Factor and Factoring by Grouping
Learning Objectives
By the end of this section, you should be able to:
- Find the greatest common factor (GCF) of two or more terms.
- Factor the GCF from a polynomial.
- Factor a polynomial by grouping.
Finding the Greatest Common Factor
Every polynomial is made up of smaller terms.
The greatest common factor (GCF) is the largest quantity—number, variable, or both—that divides each term exactly.
Example 5.1.1
Find the GCF of 12x² and 18x³.
Solution
GCF(12, 18) = 6 and the smallest power of x is x², so the GCF is 6x².
Example 5.1.2
Find the GCF of 8a³b² and 12a²b³.
Solution
GCF(8, 12) = 4. The smallest powers are a² and b², so the GCF is 4a²b².
Try It
Factoring Out the Greatest Common Factor
Once you identify the GCF, you can factor it out by dividing each term by the GCF.
This is the reverse of distributing.
Example 5.1.3
Factor 6x² + 9x.
Solution
GCF = 3x, so:
6x² + 9x = 3x(2x + 3).
Example 5.1.4
Factor 8x³ − 4x² + 12x.
Solution
GCF = 4x, so:
8x³ − 4x² + 12x = 4x(2x² − x + 3).
Example 5.1.5
Factor 15a²b³c + 20ab²c².
Solution
GCF = 5ab²c, so:
15a²b³c + 20ab²c² = 5ab²c(3a + 4c).
Example 5.1.6
Factor −14x³ + 21x².
Solution
It’s often helpful to factor out a negative GCF so the leading term inside the parentheses is positive.
GCF = −7x², so:
−14x³ + 21x² = −7x²(2x − 3).
Example 5.1.7
Factor 5x³ − 7x² + 3x − 2.
Solution
The GCF is 1, and the polynomial does not factor further using the methods in this section.
Prime polynomial.
Try It
Factoring by Grouping
When a polynomial has four terms, it may factor by grouping.
Group the first two and the last two terms, factor each group’s GCF, then factor out the common binomial.
Example 5.1.8
Factor x²(4x − 1) − 6(4x − 1).
Solution
Both terms share (4x − 1), so:
x²(4x − 1) − 6(4x − 1) = (4x − 1)(x² − 6).
Example 5.1.9
Factor ax + ay + bx + by.
Solution
Group: (ax + ay) + (bx + by)
= a(x + y) + b(x + y)
= (x + y)(a + b).
Example 5.1.10
Factor 4x³ − 6x² − 8x² + 12x.
Solution
Group: (4x³ − 6x²) + (−8x² + 12x)
Factor each group:
= 2x²(2x − 3) − 4x(2x − 3)
Factor out the common binomial:
= (2x − 3)(2x² − 4x)
Now factor out the GCF from the remaining polynomial:
= (2x − 3)·2x(x − 2)
Final: 2x(2x − 3)(x − 2)
Important: Always check for a GCF first before using grouping.
Example 5.1.11
Factor 3x³ − 2x² − 9x + 6.
Solution
Group: (3x³ − 2x²) + (−9x + 6)
Factor GCFs:
= x²(3x − 2) − 3(3x − 2)
Factor out the common binomial:
= (3x − 2)(x² − 3).
Example 5.1.12
Factor a³ + 2a² + 3a + 6.
Solution
Group: (a³ + 2a²) + (3a + 6)
Factor each group:
= a²(a + 2) + 3(a + 2)
Factor out the common binomial:
= (a + 2)(a² + 3).
Example 5.1.13
Factor x³ + 2x² + x + 5.
Solution
Group: (x³ + 2x²) + (x + 5)
= x²(x + 2) + 1(x + 5)
There is no common binomial, so this polynomial does not factor by grouping.
Prime polynomial.
Try It
Concept Check
Key Takeaways
- The GCF is the largest factor common to all terms.
- Always factor out the GCF first before trying other methods.
- For four-term polynomials, try factoring by grouping.
- If no common factor or binomial remains, the polynomial is prime.
