Factorization by Grouping

Factorization by grouping is a method used to factorize algebraic expressions containing four or more terms.

The terms are arranged in groups. The highest common factor is removed from each group. If the same bracket remains in every group, that bracket becomes a common factor.

What You Will Learn

• Recognize expressions that can be factorized by grouping
• Arrange terms into suitable groups
• Find and remove the HCF of each group
• Identify and remove a common bracket
• Rearrange terms when the original grouping does not work
• Factorize expressions containing negative terms
• Check a factorization by expanding

Before You Begin

You should already be able to identify terms and coefficients, find the HCF of algebraic terms, remove a common factor, apply the distributive law and expand two brackets.

Key Vocabulary

Term	Meaning	Example
Term	A part separated by a + or − sign	6x and −9y
Group	Terms considered together	(ax + bx)
Common factor	A factor shared by every term in a group	x in ax + bx
Common bracket	The identical bracket produced from each group	(a + b)
Factorize	Rewrite an expression as a product	ax + ay = a(x + y)
Rearrange	Change the order of terms without changing the expression	ax + by + ay + bx becomes ax + ay + bx + by

Reviewing the Common-Factor Method

Before using grouping, recall how to factorize two terms.

Similarly:

Why Grouping Works

Factorization by grouping is another application of the distributive law. A complete bracket can act as one common algebraic factor.

In the same way:

The Main Idea

Consider the expression:

There is no single variable common to all four terms. Group the terms into pairs:

Factorize each group:

Both parts contain the common bracket (a + b):

Visual Mapping

Part	Becomes
Common bracket	(a + b)
Remaining factors	(x + y)

The General Pattern

Steps for Factorization by Grouping

1. Check whether the entire expression has an HCF.
2. Arrange the terms into two suitable groups.
3. Factorize the HCF from each group.
4. Check that the same bracket remains in both groups.
5. Remove the common bracket.
6. Place the remaining factors in a second bracket.
7. Expand the final answer to check it.

Worked Examples

Example 1: Factorize ax + bx + ay + by.

Example 2: Factorize 2ax + 2ay + 3bx + 3by.

1. Group the terms: (2ax + 2ay) + (3bx + 3by).
2. The HCF of the first group is 2a: 2a(x + y).
3. The HCF of the second group is 3b: 3b(x + y).

Example 3: Factorize 6ac + 9ad + 4bc + 6bd.

1. Group the terms: (6ac + 9ad) + (4bc + 6bd).
2. The HCF of the first group is 3a: 3a(2c + 3d).
3. The HCF of the second group is 2b: 2b(2c + 3d).

Grouping Expressions with Products

Example 4: Factorize 3x + 3y + ax + ay.

Example 5: Factorize 5x − 5y + 2ax − 2ay.

When One Group Has Coefficient 1

Example 6: Factorize 2pq + 6p + q + 3.

Grouping Expressions with Powers

Example 7: Factorize x² + 3x + 2x + 6.

Example 8: Factorize 2x² + 6x + 3x + 9.

When the Brackets Do Not Match

When the Terms Must Be Rearranged

Example 9: Factorize ax + by + ay + bx.

Grouping the terms as written does not produce identical brackets:

Rearrange the terms:

Now group and factorize:

Grouping with Negative Terms

Example 10: Factorize ax − ay + bx − by.

Example 11: Factorize x² − 4x − 3x + 12.

Why a Negative Factor May Be Needed

Consider:

The brackets are reversed, but:

First Check for an Overall HCF

Example 12: Factorize 6x² + 12x + 18x + 36 fully.

All four terms share a factor of 6:

Group the terms inside the bracket:

Should I Use Factorization by Grouping?

Check for Understanding

How to Check by Expanding

Check the factorization (x + 3)(x + 2):

This matches the original expression x² + 3x + 2x + 6, so the factorization is correct.

Quick Reference Table

Original expression	After factorizing each group	Final answer
ax + bx + ay + by	x(a + b) + y(a + b)	(a + b)(x + y)
2ax + 2ay + 3bx + 3by	2a(x + y) + 3b(x + y)	(x + y)(2a + 3b)
6ac + 9ad + 4bc + 6bd	3a(2c + 3d) + 2b(2c + 3d)	(2c + 3d)(3a + 2b)
2pq + 6p + q + 3	2p(q + 3) + 1(q + 3)	(q + 3)(2p + 1)
x2 + 3x + 2x + 6	x(x + 3) + 2(x + 3)	(x + 3)(x + 2)
x2 − 4x − 3x + 12	x(x − 4) − 3(x − 4)	(x − 4)(x − 3)

Common Mistakes

• Grouping without checking: factorizing each group must produce identical brackets.
• Removing the wrong HCF: 6x + 9y = 3(2x + 3y).
• Using different brackets: x(x + 3) + 2(x + 6) cannot be factorized by removing a common bracket.
• Forgetting coefficient 1: q + 3 = 1(q + 3).
• Losing a negative sign: −3x + 12 = −3(x − 4).
• Failing to rearrange: try placing terms with related factors together.
• Ignoring an overall HCF: remove it before grouping.
• Stopping too early: make sure no further common factor remains.

Multiple-Choice Questions

Practice Questions

Factorize each expression fully.

1. ax + ay + bx + by
2. ab + ac + db + dc
3. 2ax + 2ay + 3bx + 3by
4. 6ac + 9ad + 4bc + 6bd
5. 3x + 3y + ax + ay
6. 5x − 5y + 2ax − 2ay
7. 2pq + 6p + q + 3
8. x² + 4x + 3x + 12
9. x² + 6x + 2x + 12
10. 2x² + 8x + 3x + 12
11. 3x² + 6x + 5x + 10
12. x² − 5x − 2x + 10
13. 2x² − 6x + x − 3
14. ax + by + ay + bx
15. 6x² + 12x + 18x + 36
16. a(x − y) + b(y − x)
17. Check Question 10 by expanding.

Answers

1. (a + b)(x + y)
2. (a + d)(b + c)
3. (x + y)(2a + 3b)
4. (2c + 3d)(3a + 2b)
5. (x + y)(a + 3)
6. (x − y)(5 + 2a)
7. (q + 3)(2p + 1)
8. (x + 4)(x + 3)
9. (x + 6)(x + 2)
10. (x + 4)(2x + 3)
11. (x + 2)(3x + 5)
12. (x − 5)(x − 2)
13. (x − 3)(2x + 1)
14. (a + b)(x + y)
15. 6(x + 2)(x + 3)
16. (x − y)(a − b)
17. (x + 4)(2x + 3) = 2x² + 3x + 8x + 12 = 2x² + 8x + 3x + 12.

Lesson Summary