Factoring by Grouping

Learn how to factor polynomials with four terms by grouping pairs that share common factors.

Intermediate25 minLesson
Printable worksheet

Definition

Factoring by grouping is a method used to factor polynomials with four terms. The key idea is to split the polynomial into two pairs and factor out the Greatest Common Factor (GCF) from each pair.
The steps are:
  1. 1.Group the first two terms and the last two terms
  2. 2.Factor out the GCF from each group
  3. 3.Factor out the common binomial factor
ax+ay+bx+by=a(x+y)+b(x+y)=(x+y)(a+b)ax + ay + bx + by = a(x + y) + b(x + y) = (x + y)(a + b)
The method works when both groups share a common binomial factor after factoring out their GCFs.

Try it now

What is the first step in factoring by grouping?

Worked Examples

Factor: x3+2x2+3x+6x^3 + 2x^2 + 3x + 6
1

Group the first two terms and last two terms

(x3+2x2)+(3x+6)(x^3 + 2x^2) + (3x + 6) → Two groups formed
2

Factor out the GCF from the first group

x2(x+2)+(3x+6)x^2(x + 2) + (3x + 6) → GCF of first group: x2x^2
3

Factor out the GCF from the second group

x2(x+2)+3(x+2)x^2(x + 2) + 3(x + 2) → GCF of second group: 33
4

Factor out the common binomial (x+2)(x + 2)

(x+2)(x2+3)(x + 2)(x^2 + 3) → Final factored form

Common Mistakes

Not finding the correct GCF for each group

Why it's wrong: If the GCF is incorrect, the binomials won't match and you can't complete the factoring.

Correct: Always find the greatest common factor, not just any common factor. For 6x2+9x6x^2 + 9x, the GCF is 3x3x, not just 33.

Forgetting to factor out a negative when needed

Why it's wrong: Sometimes you need to factor out a negative GCF to make the binomials match.

Correct: For −2x−4-2x - 4, factor out −2-2 to get −2(x+2)-2(x + 2), which matches (x+2)(x + 2) from the other group.

Groups don't have a common binomial factor

Why it's wrong: This usually means the terms were grouped incorrectly or the polynomial cannot be factored by grouping.

Correct: Try rearranging the terms before grouping. If no arrangement works, the polynomial may not factor by grouping.

Dropping the second factor when writing the final answer

Why it's wrong: Students sometimes write only the common binomial and forget the remaining factor.

Correct: The answer has two factors: the common binomial AND the factor formed by the GCFs.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Factoring Polynomials by Grouping | Step-by-Step Method with Examples

by Understand The Math

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Interactive Visual

Function graph

-4-3-2-10123-20-10010203040xy(-2, 0)

Tap or hover over the graph to read off values.

x^3 + 2x^2 + 3x + 6(x + 2)(x^2 + 3)
  • Zero (x-intercept): (-2, 0)
  • Zero (x-intercept): (-2, 0)

Table of values

x-2-1012
x^3 + 2x^2 + 3x + 60461228
(x + 2)(x^2 + 3)0461228

Interactive Sandbox

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is the first step in factoring by grouping?

Why It Matters

Factoring by grouping is essential for:
  • Solving polynomial equations: Many equations require factoring before you can find solutions
  • Simplifying expressions: Complex algebraic expressions become easier to work with when factored
  • Advanced mathematics: This technique is a foundation for factoring trinomials using the AC method
  • Real applications: Engineers and scientists use factoring to solve problems in physics, economics, and computer science
Mastering this skill prepares you for more advanced algebra topics like factoring trinomials and solving quadratic equations.

Real World Applications

Engineering and Design

Engineers factor polynomials when solving structural equations and optimizing designs.

Example:

When calculating the dimensions of a beam that can support a load, engineers use factored polynomials to find critical points.

1Try It Yourself

An engineer is solving the equation x3+3x2+2x+6=0x^3 + 3x^2 + 2x + 6 = 0 to find design parameters.

Factor the left side to help solve the equation.

Step 1: Write the mathematical expression

Group and factor: (x3+3x2)+(2x+6)(x^3 + 3x^2) + (2x + 6)

Economics and Business

Economists use polynomial factoring to analyze cost functions and find break-even points.

Example:

A profit function P(x)=2x3−6x2+4x−12P(x) = 2x^3 - 6x^2 + 4x - 12 can be factored to find when profits equal zero.

2Try It Yourself

Factor 2x3−6x2+4x−122x^3 - 6x^2 + 4x - 12 to analyze the profit function.

What is the factored form?

Step 1: Write the mathematical expression

Group: (2x3−6x2)+(4x−12)(2x^3 - 6x^2) + (4x - 12)

Key Takeaways

  • 1Factoring by grouping works best on polynomials with four terms
  • 2Group terms in pairs and factor out the GCF from each pair
  • 3Both groups must share a common binomial factor
  • 4Sometimes you need to factor out a negative GCF to make binomials match
  • 5Always verify your answer by multiplying the factors back together

Frequently Asked Questions

Try grouping the terms and factoring each group. If both groups share a common binomial factor after factoring out their GCFs, the method works. If not, try rearranging terms or use a different factoring method.
Try grouping the terms and factoring each group. If both groups share a common binomial factor after factoring out their GCFs, the method works. If not, try rearranging terms or use a different factoring method.
Try factoring out a negative GCF from one group. For example, instead of factoring out 2 from −2x−4-2x - 4, factor out −2-2 to get −2(x+2)-2(x + 2), which might match the other binomial.
Sometimes yes. If grouping the first two and last two terms doesn't work, try grouping the first and third terms together. The goal is to create groups with a common binomial factor.

Glossary

Factoring by grouping
A method of factoring four-term polynomials by grouping terms in pairs and finding common factors
Greatest Common Factor (GCF)
The largest factor that divides evenly into all terms of a group
Binomial factor
An algebraic expression with two terms that appears as a factor in the polynomial
Common binomial
A binomial expression that appears in both groups after factoring out the GCFs
Today's challengeOne problem, three tries

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