Factorisation Calculator

Use ^ for powers (x^2), * for multiplication (2*x), + for addition, – for subtraction

Factorisation Methods

Factorisation is the process of expressing an algebraic expression as a product of its factors. Our calculator supports various factorisation methods for different types of polynomials and expressions.

Greatest Common Factor (GCF)

Find the largest factor common to all terms and factor it out.

Example: 6x + 12 = 6(x + 2)

Quadratic Trinomials

Factor expressions of the form ax² + bx + c by finding two numbers that multiply to ac and add to b.

Example: x² + 5x + 6 = (x + 2)(x + 3)

Difference of Squares

Use the formula a² – b² = (a + b)(a – b) for expressions with two perfect squares being subtracted.

Example: x² – 16 = (x + 4)(x – 4)

Sum and Difference of Cubes

Apply special formulas for cubic expressions involving addition or subtraction of perfect cubes.

Example: x³ + 8 = (x + 2)(x² – 2x + 4)

Special Factorisation Formulas

Difference of Squares

a² – b² = (a + b)(a – b)

This formula applies when you have two perfect squares separated by subtraction. Remember that the sum of squares (a² + b²) cannot be factored over real numbers.

Sum of Cubes

a³ + b³ = (a + b)(a² – ab + b²)

Notice the pattern: the first factor is the sum of the cube roots, and the second factor has alternating signs.

Difference of Cubes

a³ – b³ = (a – b)(a² + ab + b²)

Similar to the sum of cubes, but the first factor shows subtraction and the second factor has all positive terms.

Perfect Square Trinomials

a² + 2ab + b² = (a + b)²
a² – 2ab + b² = (a – b)²

These patterns occur when the middle term is twice the product of the square roots of the first and last terms.

Step-by-Step Factorisation Guide

For Quadratic Expressions (ax² + bx + c)

  1. Check for GCF: Look for common factors in all terms first
  2. Identify coefficients: Note the values of a, b, and c
  3. Find factor pairs: Look for two numbers that multiply to give ac and add to give b
  4. Split the middle term: Rewrite bx using the two numbers found
  5. Factor by grouping: Group terms in pairs and factor each group
  6. Extract common binomial: Factor out the common binomial factor

For Higher-Degree Polynomials

  1. Look for patterns: Check if it matches special forms (sum/difference of cubes, etc.)
  2. Try grouping: Group terms strategically to find common factors
  3. Use substitution: For complex expressions, try substituting parts with single variables
  4. Factor completely: Continue factoring until no further factorisation is possible

Common Factorisation Examples

Simple Quadratic:
Expression: x² + 7x + 12
Solution: Look for two numbers that multiply to 12 and add to 7
Numbers: 3 and 4 (3 × 4 = 12, 3 + 4 = 7)
Answer: (x + 3)(x + 4)
Difference of Squares:
Expression: 4x² – 25
Solution: Recognise as (2x)² – 5²
Answer: (2x + 5)(2x – 5)
Trinomial with Leading Coefficient:
Expression: 3x² + 10x + 8
Solution: Find factors of 3 × 8 = 24 that add to 10
Numbers: 6 and 4, so 10x = 6x + 4x
Rewrite: 3x² + 6x + 4x + 8 = 3x(x + 2) + 4(x + 2)
Answer: (x + 2)(3x + 4)
Sum of Cubes:
Expression: 27x³ + 64
Solution: Recognise as (3x)³ + 4³
Answer: (3x + 4)(9x² – 12x + 16)

Frequently Asked Questions

What is factorisation in mathematics?
Factorisation is the process of breaking down a mathematical expression into a product of simpler expressions called factors. It’s essentially the reverse of expanding brackets. For example, x² + 5x + 6 can be factorised as (x + 2)(x + 3).
How do I know which factorisation method to use?
Start by looking for a greatest common factor (GCF) first. Then identify the type of expression: quadratic trinomials use the ac method, difference of squares uses a² – b² = (a+b)(a-b), and sum/difference of cubes have their specific formulas. Practice helps you recognise patterns quickly.
Can all quadratic expressions be factorised?
Not all quadratics can be factorised using real numbers. If the discriminant (b² – 4ac) is negative, the quadratic has no real factors. However, it can still be factorised using complex numbers or expressed in completed square form.
What’s the difference between factoring and factorising?
These terms are used interchangeably in mathematics. “Factoring” is more common in American English, whilst “factorising” is preferred in British English. Both refer to the same mathematical process of expressing an expression as a product of factors.
Why is factorisation important?
Factorisation is crucial for solving equations, simplifying expressions, finding roots of polynomials, and solving real-world problems. It’s fundamental to algebra and appears frequently in calculus, where it helps with integration and differentiation of complex expressions.
How can I check if my factorisation is correct?
Multiply out your factors using FOIL (First, Outer, Inner, Last) or distribution. If you get back to the original expression, your factorisation is correct. This is always a good practice to verify your work.

Tips for Successful Factorisation

Recognition Patterns

  • Perfect squares: Look for expressions where the first and last terms are perfect squares
  • Common factors: Always check if all terms share a common factor before applying other methods
  • Sign patterns: Pay attention to positive and negative signs, especially in difference of squares
  • Coefficient relationships: In trinomials, check if the middle coefficient relates to the other terms

Problem-Solving Strategies

  • Start with the simplest approach: look for common factors first
  • Write expressions in standard form (descending powers) before factoring
  • Use the discriminant to determine if a quadratic can be factorised over real numbers
  • Practice recognising special patterns like perfect square trinomials
  • Don’t forget to factor completely – some expressions may require multiple steps

Avoiding Common Mistakes

  • Remember that a² + b² cannot be factorised over real numbers (it’s not a difference of squares)
  • Be careful with signs when factoring – check your work by expanding
  • Don’t forget to factor out the GCF first, especially in more complex expressions
  • Ensure all factors are completely factorised – sometimes factors can be factorised further

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