1. Introduction
Sometimes an expression does not have one common factor for all terms. In such cases, we use factorisation by grouping. This method works especially well for four-term expressions.
The idea is simple: group terms in pairs, take out a common factor from each pair, and then factorise the remaining common binomial factor.
2. When Do We Use Grouping?
Grouping is used when:
- There is no single common factor for the entire expression.
- The expression has 4 terms or can be rearranged to look like it.
- We notice patterns such as:
\(ab + ac + bd + cd\) - Two pairs of terms share separate common factors.
3. Steps for Factorising by Grouping
- Divide the expression into two groups.
- Take the common factor out from each group.
- Check if a common binomial factor appears.
- Factor out the binomial factor.
- Write final answer as:
common binomial × common factor expression
3.1. Example Walkthrough 1 (Standard Pattern)
Factorise: \(ax + ay + bx + by\)
Group terms:
\((ax + ay) + (bx + by)\)
Take common factor from each group:
- From \(ax + ay\): take out \(a\) → \(a(x + y)\)
- From \(bx + by\): take out \(b\) → \(b(x + y)\)
Now expression becomes:
\(a(x + y) + b(x + y)\)
Take out \((x + y)\):
\((x + y)(a + b)\)
3.2. Example Walkthrough 2 (Rearrangement Needed)
Factorise: \(3x + 6y - 2y - x\)
Rearrange terms:
\(3x - x + 6y - 2y\)
Group them:
\((3x - x) + (6y - 2y)\)
Take common factors:
- From \(3x - x\): take \(x\) → \(x(3 - 1)\)
- From \(6y - 2y\): take \(2y\) → \(2y(3 - 1)\)
Expression becomes:
\(x(2) + 2y(2)\)
Take out 2:
\(2(x + y)\)
4. More Examples
1. Factorise: \(2a + 2b + ax + bx\)
Group → \((2a + 2b) + (ax + bx)\)
Common factors → \(2(a + b) + x(a + b)\)
Final Answer → \((a + b)(2 + x)\)
2. Factorise: \(pq + pr - q - r\)
Group → \((pq + pr) - (q + r)\)
Common factors → \(p(q + r) - 1(q + r)\)
Final Answer → \((q + r)(p - 1)\)
3. Factorise: \(x^2 + 3x + 2x + 6\)
Group → \((x^2 + 3x) + (2x + 6)\)
Common → \(x(x + 3) + 2(x + 3)\)
Final Answer → \((x + 3)(x + 2)\)
5. Common Mistakes
- Grouping incorrectly so that no common binomial appears.
- Forgetting to rearrange terms when needed.
- Not taking the highest common factor from each group.
- Mistakenly pairing wrong terms.
6. Quick Practice
Factorise:
- \(ab + ac + db + dc\)
- \(x^2 - 5x + 2x - 10\)
- \(3p + 6q + p^2 + 2pq\)
- \(ax - bx + ay - by\)
7. Summary
- Grouping is used when no single common factor exists for all terms.
- Works well for four-term expressions.
- Goal is to make a common binomial appear.
- Final expression is always written as the product of two factors.