Factorising sits at the heart of algebra and number theory. At its simplest, it is the process of rewriting an expression as a product of its factors. For numbers, factorising reveals prime components; for algebraic expressions, it uncovers hidden structure that makes equations easier to solve. In British maths classrooms, you may hear the question phrased as what is factorising and then shown a series of techniques to break expressions apart. This article provides a clear, step‑by‑step guide to both numerical and algebraic factorising, with plenty of examples, explanations, and practical tips.
What is Factorising? A Clear Definition
Broadly speaking, factorising means expressing a mathematical object as a product of its factors. For integers, that means writing a number as a multiplication of prime numbers. For polynomials, it means writing the polynomial as a product of simpler polynomials. The idea is that the factors multiply together to give the original expression, and that the factors are as simple or as fundamental as possible.
When people ask What is Factorising?, they usually want to know two things: the methods used to factorise, and the ways those methods help with solving problems. In many cases factorising is the key to solving equations, simplifying expressions, or proving identities. It is also a powerful tool in areas such as number theory, algebra, and even calculus, where factoring can simplify integration or differentiation in certain contexts.
Factorising Numbers: Prime Factorisation
Factorising numbers is the most tangible form of the idea. The goal is to express a whole number as a product of prime numbers. This is called prime factorisation. Every integer greater than 1 has a unique prime factorisation (up to the order of the factors), a result known as the Fundamental Theorem of Arithmetic.
Prime Factorisation: The Basics
To factorise a number, you start with the smallest prime and divide as far as possible. For example, factorising 60: 60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5. So the prime factorisation of 60 is 2² × 3 × 5.
- Understand that prime numbers have exactly two distinct positive divisors: 1 and themselves. 2, 3, 5, 7, 11, and so on.
- Continue dividing by primes until the remaining quotient is 1.
- Write the product of the primes with their multiplicities (exponents) where needed.
Examples to Try
Try factorising these numbers using prime factorisation:
- 84 = 2² × 3 × 7
- 210 = 2 × 3 × 5 × 7
- 108 = 2² × 3³
Prime factorisation is not only a tool in itself; it also helps with simplifying fractions, finding greatest common factors, and determining least common multiples. When you know the prime factors, you can compare numbers at a granular level to see common building blocks.
Factorising Algebraic Expressions: The Core Techniques
Factorising algebraic expressions is a different kind of challenge from numerical factorisation. Here, your job is to rewrite a polynomial as a product of simpler polynomials, such as binomials or monomials. The key idea is to identify common factors, patterns, and combinations that re-create the original expression when multiplied.
Common Factors: The First Step
Always start by checking for a common factor. If every term in the polynomial shares a common factor, factor it out. For example, factorising the expression 6x² + 9x can be done by extracting the greatest common factor (GCF), which is 3x: 6x² + 9x = 3x(2x + 3).
Factoring Quadratic Trinomials
One of the most common factorising tasks in algebra is dealing with quadratics of the form ax² + bx + c. The standard method is to find two numbers that multiply to ac and add to b. Then split the middle term and factor by grouping. For instance, factorise x² + 5x + 6:
- ac = 1 × 6 = 6. Look for two numbers that multiply to 6 and add to 5: 2 and 3.
- Rewrite as x² + 2x + 3x + 6 and factor by grouping: x(x + 2) + 3(x + 2) = (x + 3)(x + 2).
When a = 1, this becomes straightforward. For ax² + bx + c with a ≠ 1, you apply the ac method, or sometimes a substitution to simplify the process. Practice with examples such as 2x² + 7x + 3, which factors to (2x + 1)(x + 3).
Difference of Squares
The difference of squares is a special, highly useful factorising pattern. If you have an expression of the form a² − b², it factors as (a − b)(a + b). For example, x² − 9 = (x − 3)(x + 3). Recognising this pattern can turn a complex expression into a simple product very quickly.
Factorising by Grouping
When you have four or more terms, factorising by grouping can be effective. A classic case is when you can rearrange terms to reveal common binomial factors. For example, factorise expression x³ + 3x² + 2x + 6 by grouping: (x³ + 3x²) + (2x + 6) = x²(x + 3) + 2(x + 3) = (x² + 2)(x + 3).
Special Products and Reverse Patterns
Some expressions are built from special products like the square of a binomial or the sum/difference of cubes. Remember:
- Square of a binomial: a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)²
- Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²)
- Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²)
Recognising these patterns expands your ability to factorise quickly and accurately.
Factoring Trigonometric, Exponential, and Polynomial Expressions
Factorising isn’t limited to simple polynomials. In higher maths, you may deal with expressions that involve second or third degree polynomials in a variable, or products of linear and quadratic factors. The same principles apply: search for common factors, apply the difference of squares or special products, and use grouping as necessary. Practice with expressions like 3x² − 12x + 12 or x⁴ − 5x² + 6 to see how these ideas translate beyond the simple cases.
Why Factorising Matters in Maths
Factorising is a foundational skill for solving equations, simplifying expressions, and understanding the structure of mathematical objects. Here are some key reasons why what is factorising matters in practice:
- Solving equations: Factorised forms often reveal the roots directly. For example, if f(x) = (x − 2)(x + 5), the zeros are x = 2 and x = −5 without needing to apply the quadratic formula.
- Rational expressions: To simplify a fraction, factor the numerator and denominator and cancel common factors. See how what is factorising helps to make complex fractions manageable.
- Arithmetic and number theory: Prime factorisation is essential for working with divisibility, greatest common factors, and least common multiples.
- Polynomial identity and rearrangement: Factorising can lead to elegant proofs and compact representations of expressions.
- Calculus preparation: In certain cases, integrating rational functions or differentiating products becomes easier after factoring.
Step-by-Step Examples: From Simple to Sophisticated
Working through concrete examples helps solidify the concept of what is factorising. Here are a few progressively challenging scenarios.
Example 1: Factorising a Simple Monomial
Factorise 12x. The task here is to extract the greatest common factor: 12x = 12 × x, so the factorised form is 12x only if you are seeking a factorising of a constant times x. More usefully, factorise 12x² by pulling out the GCF: 12x² = 12x × x, so the factorised form is 12x(x).
Example 2: Factorising a Polynomial by GCF
Factorise 9a²b + 6ab². The GCF is 3ab, giving 3ab(3a + 2b).
Example 3: Quadratic Trinomial with a ≠ 1
Factorise 6x² + 11x + 3. We look for two numbers that multiply to 18 (ac) and add to 11 (b). Those numbers are 9 and 2. Rewrite and factor by grouping: 6x² + 9x + 2x + 3 = 3x(2x + 3) + 1(2x + 3) = (2x + 3)(3x + 1).
Example 4: Difference of Squares
Factorise x² − 16. This is a difference of squares: (x − 4)(x + 4).
Example 5: Factorising by Grouping
Factorise x³ + 3x² + 2x + 6. Grouping reveals common factors: (x³ + 3x²) + (2x + 6) = x²(x + 3) + 2(x + 3) = (x² + 2)(x + 3).
Common Mistakes in Factorising
Even experienced students can trip over factorising if they rush or misapply a rule. Here are frequent pitfalls to avoid:
- Overlooking a common factor when one is present, especially in expressions with multiple terms.
- Misapplying the ac method in quadratic factorisation; ensure you identify ac correctly and test factor pairs thoroughly.
- Assuming a factor exists because the expression looks simple; always verify by expanding the factors to recover the original expression.
- Forgetting to factorise completely; sometimes an expression factors into a product of more than two factors or requires a combination of methods.
- Neglecting the difference of squares and other special patterns when the form appears—recognising patterns saves time and reduces errors.
Practice Tips and Strategies
To improve at factorising, adopt a systematic approach and build a toolkit of reliable strategies. Here are practical tips to develop proficiency in what is factorising:
- Start with the simplest method: always check for a common factor first. It’s the quickest win and can simplify the problem dramatically.
- When dealing with quadratics, practise the ac method and the factoring by grouping with various examples to become fluent in recognising patterns.
- Write out the factorised form first, then verify by expansion. This helps you catch mistakes early.
- Keep a mental or written list of standard patterns: difference of squares, perfect square trinomials, sum/difference of cubes, and common polynomial identities.
- Use regular spaced practice with increasingly complex expressions to improve fluency and speed.
How to Check Your Factorisations
Verification is an essential step. After factorising, expand the factors to confirm you recover the original expression. For example, if you propose that x² + 5x + 6 = (x + 2)(x + 3), expand to check: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6, which matches the original. This simple check can save time and frustration, especially with more complex expressions.
Real-World Applications of Factorising
While factorising is a theoretical skill, it has practical uses in everyday maths and beyond:
- Engineering and physics sometimes require factoring polynomials to simplify model equations or to identify resonant modes and critical points.
- Economics and statistics may involve factoring rational expressions during data modelling or in solving optimisation problems.
- Computer science uses factoring ideas in algorithms related to polynomial time computations and in coding theory.
- Everyday maths — learning to factor makes it easier to understand algebraic manipulations encountered in coursework and exams.
Terminology and Quick Reference
Below is a concise glossary to help you navigate common factorising terms and concepts. This quick reference supports understanding of what is factorising in different contexts:
- Factor: A number or expression that divides another exactly; in algebra, a factor is a polynomial that multiplies with another to give the original expression.
- Common factor/GCF: The greatest common factor shared by all terms in an expression; extracting it is often the first step in factorising.
- Prime factorisation: Expressing a number as a product of prime numbers.
- Quadratic trinomial: A polynomial of the form ax² + bx + c; often factorised into two binomials when possible.
- Difference of squares: A² − B² = (A − B)(A + B); a frequent special case of factorising.
- Factoring by grouping: A method used for polynomials with four or more terms, organised to reveal common binomial factors.
- AC method: A technique for factoring quadratics where you split the middle term using two numbers that multiply to ac and add to b.
FAQ: What is Factorising? Quick Answers
Here are a few frequently asked questions related to what is factorising and how it is used in practice:
- Q: What is Factorising in simple terms? A: It is rewriting an expression as a product of its factors, either numbers or polynomials, to reveal structure and simplify problems.
- Q: Why is factorising important in algebra? A: It helps solve equations faster, simplifies expressions, and exposes roots or zeroes of polynomials.
- Q: How do I know when to use the difference of squares? A: If you see a term that is a perfect square subtracted from another perfect square, you likely have a difference of squares pattern.
- Q: Can every polynomial be factorised? A: Not always. Some polynomials are irreducible over the integers or the rationals, but may factor over the reals or complex numbers.
Conclusion: Mastering What is Factorising
What is factorising is a foundational idea that unlocks many strands of mathematics. By understanding both the numerical aspect—prime factorisation—and the algebraic side—factoring polynomials—you gain a powerful tool for problem solving, proving identities, and simplifying complex expressions. With practice, the patterns become second nature, and what is factorising shifts from a daunting task to a straightforward, strategic approach. Whether you are preparing for exams, tackling high‑level maths at university, or simply aiming to sharpen your mathematical thinking, mastering factorising will pay dividends across countless areas of study and work.
Additional Practice and Resources
To deepen your understanding of what is factorising, consider incorporating the following practices into your study routine:
- Regularly work through factorising problems of increasing difficulty, keeping track of which methods work best in different contexts.
- Keep a personal notebook of factorising tricks, including a catalogue of patterns and examples that you can revisit quickly during study sessions.
- Try a weekly challenge: pick a polynomial and attempt to factorise it using at least two different methods, then compare results and discuss why one approach may be more efficient.
- Use online quizzes and printable worksheets that focus specifically on factorising to reinforce learning outside the classroom.
In the end, what is factorising? It is the art of revealing hidden structure. It is the technique of turning a single expression into a product of several simpler parts, each contributing to a clearer understanding of the whole. With patience, practice, and the right strategies, factorising becomes a natural and reliable tool in your mathematical toolkit.