Free Factorisation Calculator
Factorise quadratics, differences of squares, sum and difference of cubes, higher-degree polynomials, expand brackets, and complete the square — with full step-by-step working, KaTeX rendering, and interactive Learn Mode.
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What is Factorisation?
Factorisation is the process of writing an algebraic expression as a product of its factors. It is the reverse of expanding brackets. For example, x² + 5x + 6 factorises to (x + 2)(x + 3) because (x + 2)(x + 3) expands back to x² + 5x + 6.
Expanding vs Factorising
Expanding (multiplying out):
(x + 2)(x + 3) → x² + 5x + 6
Factorising (reverse):
x² + 5x + 6 → (x + 2)(x + 3)
Factorisation is a core exam skill that appears throughout GCSE and A-Level Maths. It underpins solving quadratic equations, simplifying algebraic fractions, and is essential groundwork for A-Level calculus and integration.
Step-by-Step: Simple Quadratics (a = 1)
When the coefficient of x² is 1, find two numbers that add to b and multiply to c.
Write the quadratic
x² + 7x + 12
Identify b and c
b = 7, c = 12
Find the factor pair of c
Pairs of 12: (1,12), (2,6), (3,4)
Check which pair adds to b
3 + 4 = 7 ✓
Write the factorised form
(x + 3)(x + 4)
Sign Rules for (x + p)(x + q)
When c > 0 and b > 0:
Both p and q are positive
x² + 5x + 6 = (x+2)(x+3)
When c > 0 and b < 0:
Both p and q are negative
x² − 5x + 6 = (x−2)(x−3)
When c < 0:
p and q have opposite signs
x² + x − 6 = (x+3)(x−2)
Step-by-Step: The AC Method (a ≠ 1)
When the leading coefficient a ≠ 1, multiply a × c first, then find a pair that adds to b.
Worked Example: 2x² + 7x + 3
= (x + 3)(2x + 1) ✓
The Difference of Two Squares
a² − b² = (a + b)(a − b)
Both terms must be perfect squares, and the operation must be subtraction. The sum of squares a² + b² cannot be factorised over ℝ.
Geometric Proof
Take an a×a square and remove a b×b square from one corner. The L-shaped remainder (area = a²−b²) can be rearranged into a rectangle of dimensions (a+b) × (a−b) — proving the identity geometrically.
Examples:
Sum & Difference of Cubes — SOAP Method
The SOAP Mnemonic
Sum of cubes:
a³ + b³ = (a + b)(a² − ab + b²)
x³ + 8 = (x + 2)(x² − 2x + 4)
Difference of cubes:
a³ − b³ = (a − b)(a² + ab + b²)
x³ − 27 = (x − 3)(x² + 3x + 9)
Common Factor Extraction
The first step in any factorisation is to check for a Highest Common Factor (HCF) among all terms. Extract it before applying any other method.
How to Find the HCF
- 1.List all factors of each coefficient
- 2.Identify the largest number that divides all of them
- 3.For variable terms, take the lowest power common to all terms
- 4.Combine: HCF = numerical GCF × variable GCF
- 5.Divide every term by the HCF and write it outside brackets
Common Mistakes to Avoid
Not checking for GCF first
2x² + 6x + 4 = (2x + 2)(x + 2)
2x² + 6x + 4 = 2(x² + 3x + 2) = 2(x+1)(x+2)
Thinking a² + b² can be factorised
x² + 9 = (x + 3)(x + 3)
x² + 9 cannot be factorised over ℝ (sum of squares)
Wrong sign in difference of squares
x² − 9 = (x − 3)(x − 3)
x² − 9 = (x + 3)(x − 3)
SOAP sign errors for cubes
x³ + 8 = (x + 2)(x² + 2x + 4)
x³ + 8 = (x + 2)(x² − 2x + 4) — middle sign is OPPOSITE
Partial factorisation
2x² + 8x + 6 = 2(x² + 4x + 3)
2x² + 8x + 6 = 2(x + 1)(x + 3) — factorise the bracket too!
Not verifying the answer
Just write the answer without checking
Always expand back: (x+2)(x+3) = x² + 5x + 6 ✓
How Factorisation Connects to Solving Equations
Factorising a quadratic directly gives you the solutions to the equation ax² + bx + c = 0. By the zero product property, if AB = 0 then A = 0 or B = 0.
Solve: x² + 5x + 6 = 0
(x + 2)(x + 3) = 0
x + 2 = 0 → x = −2
x + 3 = 0 → x = −3
Solutions: x = −2 or x = −3
Roots = x-intercepts
The solutions of ax² + bx + c = 0 are the x-intercepts (roots) of the parabola y = ax² + bx + c. Each linear factor (x + p) gives a root at x = −p, where the parabola crosses the x-axis.
Discriminant Connection
The discriminant Δ = b² − 4ac determines factorability: Δ > 0 (perfect square) → two rational roots, Δ = 0 → one repeated root (perfect square trinomial), Δ < 0 → no real roots.
Quick Reference
Perfect Squares (1² to 15²)
1
1²
4
2²
9
3²
16
4²
25
5²
36
6²
49
7²
64
8²
81
9²
100
10²
121
11²
144
12²
169
13²
196
14²
225
15²
Perfect Cubes (1³ to 10³)
1
1³
8
2³
27
3³
64
4³
125
5³
216
6³
343
7³
512
8³
729
9³
1000
10³
More Tools: Expand, Complete the Square & Higher Degree
Expand Brackets
The reverse of factorisation — multiply out brackets using the FOIL method. Enter any factored form like (x+2)(x+3) and see the expanded polynomial with full working.
(x+2)(x+3) → x² + 5x + 6
Completing the Square
Convert any quadratic into vertex form a(x−h)² + k. Essential for finding the turning point and for sketching parabolas. Includes a geometric area visualisation.
x² + 6x + 5 → (x + 3)² − 4
Higher-Degree Polynomials
Factorise cubics, quartics and beyond using the Rational Root Theorem and synthetic division. See a factorisation tree showing the recursive decomposition.
x³ − 6x² + 11x − 6 → (x−1)(x−2)(x−3)
Frequently Asked Questions
What is factorisation?
Factorisation is writing an expression as a product of factors — the reverse of expanding. For example, x² + 5x + 6 = (x + 2)(x + 3).
How do I factorise when a = 1?
Find two numbers p and q that add to b and multiply to c, then write (x + p)(x + q). Example: x² + 7x + 12 = (x + 3)(x + 4).
What is the AC method?
For ax² + bx + c with a ≠ 1: multiply a × c, find two numbers that add to b and multiply to ac, split the middle term, group, and factor.
When should I look for a common factor first?
Always check for a GCF before anything else. Extracting it first simplifies the remaining expression and is the most commonly missed step in exams.
What is the difference between factorising and expanding?
They are inverse operations. Expanding: (x+2)(x+3) → x²+5x+6. Factorising: x²+5x+6 → (x+2)(x+3).
How do I know if a quadratic can be factorised?
Calculate Δ = b² − 4ac. If Δ is a non-negative perfect square, it can be factorised with integers. If Δ < 0, it has no real roots.
What is the discriminant?
Δ = b² − 4ac. Δ > 0: two distinct roots (factorisable if perfect square). Δ = 0: repeated root. Δ < 0: no real roots.
How do you factorise sum of cubes?
a³ + b³ = (a + b)(a² − ab + b²). Use SOAP: Same sign, Opposite sign, Always Positive.
Can a² + b² be factorised?
No — the sum of two squares cannot be factorised over the real numbers. Only a² − b² (difference) can be factorised as (a+b)(a−b).
How does factorisation relate to solving equations?
If (x + p)(x + q) = 0, then x = −p or x = −q by the zero product property. Factorisation turns a quadratic equation into two linear ones.
What is the geometric meaning of factorisation?
The factors give the x-intercepts of the parabola y = ax² + bx + c. Each factor (x + p) gives a root at x = −p where the graph crosses the x-axis.
What is completing the square?
Completing the square rewrites ax² + bx + c as a(x − h)² + k (vertex form). Take half the coefficient of x, square it, add and subtract. This reveals the vertex (h, k) of the parabola.
How does the Rational Root Theorem work?
For a polynomial with integer coefficients, all rational roots must be of the form ±p/q where p divides the constant term and q divides the leading coefficient. Test each candidate with synthetic division.
Is this calculator aligned with GCSE and A-Level syllabuses?
Yes. It covers all factorisation methods in Edexcel, AQA, and OCR syllabuses for GCSE and A-Level mathematics, including completing the square and higher-degree polynomials for A-Level.
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