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Factoring by Grouping
When four terms do not share one monomial, try grouping. Pair them so each pair has its own common factor, and hope the two brackets that remain are the same. The pattern (a + b)(c + d) expands to ac + ad + bc + bd, so grouping is that expansion in reverse. If the pairs give matching brackets, you have factored.
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Factoring by Grouping
What is it?
When four terms do not share one monomial, try grouping. Pair them so each pair has its own common factor, and hope the two brackets that remain are the same. The pattern (a + b)(c + d) expands to ac + ad + bc + bd, so grouping is that expansion in reverse. If the pairs give matching brackets, you have factored.
In simple words
- When four terms do not share one monomial, try grouping.
- From ac + ad + bc + bd, group (ac + ad) + (bc + bd) = a(c + d) + b(c + d) = (a + b)(c + d).
- If a pair is minus, factor the minus with the monomial so the leftover brackets match.
- Grouping is not limited to four terms, but four is the school pattern.
Important terms
grouping: When four terms do not share one monomial, try grouping.
(a + b)(c + d): The pattern (a + b)(c + d) expands to ac + ad + bc + bd, so grouping is that expansion in reverse.
Grouping: When four terms do not share one monomial, try grouping.
identical: The leftover binomials must be identical (or exact opposites, which you fix with a minus).
regroup: You may regroup as (ac + bc) + (ad + bd) if the first pairing fails.
What you should learn
- Explain Factoring By Grouping in simple mathematical language.
- Apply two and two correctly.
- Apply watch the signs correctly.
- Use the symbols, conditions, and formulae accurately.
- Solve a short example and check the answer.
Examples
- x3 + x2 + x + 1 = (x2 + 1)(x + 1)
- ax + ay + bx + by = (a + b)(x + y)
- 2x + 2y + zx + zy = (2 + z)(x + y)
TWO AND TWO
From ac + ad + bc + bd, group (ac + ad) + (bc + bd) = a(c + d) + b(c + d) = (a + b)(c + d). The shared binomial (c + d) becomes a common factor of the two groups. You may regroup as (ac + bc) + (ad + bd) if the first pairing fails. Same product, different pairing.
WATCH THE SIGNS
If a pair is minus, factor the minus with the monomial so the leftover brackets match. Example: ax - ay - bx + by = a(x - y) - b(x - y) = (a - b)(x - y). If you factor +b from the last two terms by mistake, the brackets will not match and grouping looks 'broken' when only the sign was wrong.
Recommended for Class 9
Full school-level explanation
When four terms do not share one monomial, try grouping. Pair them so each pair has its own common factor, and hope the two brackets that remain are the same. The pattern (a + b)(c + d) expands to ac + ad + bc + bd, so grouping is that expansion in reverse. If the pairs give matching brackets, you have factored.
Split four terms into two pairs, then factor each pair:
WHAT — Core idea
When four terms do not share one monomial, try grouping. Pair them so each pair has its own common factor, and hope the two brackets that remain are the same. The pattern (a + b)(c + d) expands to ac + ad + bc + bd, so grouping is that expansion in reverse. If the pairs give matching brackets, you have factored.
WHY — Why it matters
Grouping is not limited to four terms, but four is the school pattern. If no pairing produces equal brackets, grouping is the wrong tool; try another identity. Always expand your answer to check.
HOW — Method and examples
1. TWO AND TWO: From ac + ad + bc + bd, group (ac + ad) + (bc + bd) = a(c + d) + b(c + d) = (a + b)(c + d). 2. WATCH THE SIGNS: If a pair is minus, factor the minus with the monomial so the leftover brackets match.
HOW — Answer in an exam
The leftover binomials must be identical (or exact opposites, which you fix with a minus). If they differ, regroup or take a common minus. Do not invent a common factor that only two of four terms share and then stop.
TWO AND TWO
From ac + ad + bc + bd, group (ac + ad) + (bc + bd) = a(c + d) + b(c + d) = (a + b)(c + d). The shared binomial (c + d) becomes a common factor of the two groups. You may regroup as (ac + bc) + (ad + bd) if the first pairing fails. Same product, different pairing.
Remember
- (a + b)(c + d)
Examples
- x3 + x2 + x + 1 = (x2 + 1)(x + 1)
- ax + ay + bx + by = (a + b)(x + y)
- 2x + 2y + zx + zy = (2 + z)(x + y)
WATCH THE SIGNS
If a pair is minus, factor the minus with the monomial so the leftover brackets match. Example: ax - ay - bx + by = a(x - y) - b(x - y) = (a - b)(x - y). If you factor +b from the last two terms by mistake, the brackets will not match and grouping looks 'broken' when only the sign was wrong.
Remember
- grouping
Examples
- ax - ay - bx + by = (a - b)(x - y)
- x3 - x2 + x - 1 = (x2 + 1)(x - 1)
- 6xy - 3x + 4y - 2 = (3x + 2)(2y - 1)
Remember
- grouping
- (a + b)(c + d)
- Grouping
- identical
- regroup
- (a + b)(c + d)
- grouping
- grouping
How the ideas connect
Grouping is not limited to four terms, but four is the school pattern. If no pairing produces equal brackets, grouping is the wrong tool; try another identity. Always expand your answer to check.
Deeper understanding
Go beyond memorising Factoring by Grouping. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Grouping is not limited to four terms, but four is the school pattern. If no pairing produces equal brackets, grouping is the wrong tool; try another identity. Always expand your answer to check.
Common mistakes
Exam focus
WHAT
Key Points
HOW
Examples
Exam focus
Quick revision
- grouping
- (a + b)(c + d)
- Grouping
- identical
- grouping: When four terms do not share one monomial, try grouping.
- (a + b)(c + d): The pattern (a + b)(c + d) expands to ac + ad + bc + bd, so grouping is that expansion in reverse.
The leftover binomials must be identical (or exact opposites, which you fix with a minus). If they differ, regroup or take a common minus. Do not invent a common factor that only two of four terms share and then stop.
For long answers on Factoring by Grouping: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- grouping
- (a + b)(c + d)
- Grouping
- identical
Important formulas
- grouping
- (a + b)(c + d)
- Grouping
Common exam traps
- The leftover binomials must be identical (or exact opposites, which you fix with a minus).
- If they differ, regroup or take a common minus.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
The leftover binomials must be identical (or exact opposites, which you fix with a minus).
- Mistake
If they differ, regroup or take a common minus.
- Mistake
Correction: Do not invent a common factor that only two of four terms share and then stop.
- Mistake
Grouping is not limited to four terms, but four is the school pattern.
- Mistake
The leftover binomials must be identical (or exact opposites, which you fix with a minus). If they differ, regroup or take a common minus. Do not invent a commo
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
grouping
(a + b)(c + d)
Grouping
identical
When four terms do not share one monomial, try grouping. Pair them so each pair has its own common factor, and hope the two brackets that remain are the same. T
- Definition
- When four terms do not share one monomial, try grouping. Pair them so each pair has its own common factor, and hope the two brackets that remain are the same. T
- Example
- x3 + x2 + x + 1 = (x2 + 1)(x + 1)
- Critical warning
- The leftover binomials must be identical (or exact opposites, which you fix with a minus).
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Exam Practice
Define
Q1. Define Factoring by Grouping.
Explain
Q2. Explain: TWO AND TWO.
Example
Q3. Give one clear example of Factoring by Grouping and say why it fits.
Mistake
Q4. State one common mistake students make about Factoring by Grouping.
Exam note
Q5. Write a short exam note on Factoring by Grouping: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.