Methods of Factorisation
Methods of Factorisation
Factorisation is the reverse of expansion. It gives 2 questions of its own in the Part-III maths paper, besides being needed for solving equations and simplifying algebraic fractions. There are only four standard methods, and picking the right one on sight is the whole of the skill.
Method 1: Common Factor
Always look for a common factor first, before anything else.
6x³y − 9x²y² = 3x²y(2x − 3y). Take the highest common numerical factor and the lowest power of each common variable.
Method 2: Grouping
Used when there are four terms and no factor common to all of them. Group in pairs so that each pair yields a common factor.
ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y).
If the first grouping does not work, regroup differently before concluding the expression cannot be factorised — the pairing is not always the obvious one.
Method 3: Using Identities
| Form seen | Factorised as |
|---|---|
| a² − b² | (a + b)(a − b) |
| a² + 2ab + b² | (a + b)² |
| a² − 2ab + b² | (a − b)² |
| a³ + b³ | (a + b)(a² − ab + b²) |
| a³ − b³ | (a − b)(a² + ab + b²) |
Recognition is everything. 4x² − 25 is a difference of squares because 4x² = (2x)² and 25 = 5², so it factorises to (2x + 5)(2x − 5).
Method 4: Splitting the Middle Term
For a trinomial of the form x² + px + q, find two numbers whose sum is p and product is q.
For x² + 7x + 12: two numbers with sum 7 and product 12 are 3 and 4. So x² + 3x + 4x + 12 = x(x + 3) + 4(x + 3) = (x + 3)(x + 4).
With a coefficient on x², as in 2x² + 7x + 3, the product to match is a × c, here 2 × 3 = 6, and the sum is still 7. The numbers are 6 and 1, giving (2x + 1)(x + 3).
- If q is positive, both numbers share the sign of p.
- If q is negative, the numbers have opposite signs and the larger takes the sign of p.
- Always expand your answer mentally to check — it takes five seconds.
✗ x² − 5x + 6 = (x + 2)(x + 3) | ✓ x² − 5x + 6 = (x − 2)(x − 3) — product positive, sum negative, so both signs negative
मध्य पद को तोड़ें: दो संख्याएँ जिनका योग p और गुणनफल q हो।
TRE pointer: The sign rule in splitting the middle term is where most marks are lost, because candidates find the right pair of numbers and then attach the wrong signs. Fix the rule as positive constant means matching signs, negative constant means opposite signs. Since factorised options can be checked by expanding, this is one of the few places where verification is faster than solving — a real advantage when a wrong answer costs −1/3.
60-Second Recap
- Look for a common factor before trying anything else.
- Four terms with no common factor usually means grouping in pairs.
- Recognise a²−b², perfect squares and the sum and difference of cubes on sight.
- For x²+px+q, find two numbers with sum p and product q.
- With a coefficient on x², match the product a×c and the sum b.
- Positive constant means both signs alike; negative constant means opposite signs.
Frequently Asked Questions
How do I factorise a trinomial by splitting the middle term?
Find two numbers whose product equals the constant term and whose sum equals the coefficient of x. Split the middle term into those two, then group in pairs and take out common factors. If the x squared term has a coefficient, match the product of the first and last coefficients instead.
How do the signs work when splitting the middle term?
If the constant term is positive, both numbers carry the same sign as the middle coefficient. If it is negative, the numbers have opposite signs and the larger one takes the sign of the middle coefficient. Getting the numbers right but the signs wrong is the standard error.
When should I use grouping?
When the expression has four terms and no single factor is common to all of them. Pair the terms so each pair has a common factor, and the two resulting brackets should match. If they do not, regroup the terms differently before deciding it cannot be factorised.
Is there a quick way to check a factorisation?
Yes, expand your answer and see whether it reproduces the original expression. This takes only a few seconds and catches sign errors immediately. In a multiple-choice paper you can even expand the options rather than factorise the question, which is often faster.