How to Solve Factorisation Problems Step by Step with Examples
The concept of factorisation problems is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Factorisation methods are crucial for simplifying algebraic expressions, making complex problems easier for students from Class 7 upwards. Mastering factorisation is a key skill for board exams, Olympiads, and everyday problem solving.
Understanding Factorisation Problems
A factorisation problem asks you to rewrite an algebraic expression or a number as a product of its factors. This concept is widely used in factoring polynomials, algebraic expressions, and solving quadratic equations. Factorisation helps break complicated expressions into simpler parts, enabling you to solve or simplify equations efficiently. Whether preparing for CBSE, ICSE, or state boards, mastering different types of factorisation problems is a must.
Types of Factorisation
There are multiple methods to solve factorisation problems, including:
- Taking out the common factor
- Regrouping terms to find common factors
- Using algebraic identities, such as \( a^2 - b^2 = (a+b)(a-b) \)
- Splitting the middle term for quadratic expressions
- Prime factorisation of numbers
Each method has its use depending on the type of expression or equation given.
Stepwise Methods to Solve Factorisation Problems
Follow these steps when tackling any factorisation problem:
1. Identify and take out any common factors from all terms.2. If no common factor, check if terms can be grouped to create a common factor.
3. Look for known algebraic identities or patterns, such as the difference of squares or perfect square trinomials.
4. For quadratic expressions (like \( ax^2 + bx + c \)), split the middle term, factor each part, and group.
5. Always verify your answer by multiplying the obtained factors to check if you get the original expression.
Writing clear, step-by-step solutions helps avoid errors and builds confidence.
Worked Example – Solving a Factorisation Problem
Let’s solve a quadratic factorisation problem step-by-step:
1. Factorise \( 4x^2 + 12x + 5 \).2. Split the middle term: \( 4x^2 + 10x + 2x + 5 \).
3. Group the terms: \( (4x^2 + 10x) + (2x + 5) \).
4. Factor each group: \( 2x(2x + 5) + 1(2x + 5) \).
5. Factor out \( (2x + 5) \): \( (2x + 1)(2x + 5) \).
So, the factors of \( 4x^2 + 12x + 5 \) are \((2x + 1)(2x + 5)\). This method is used often in board exams.
More Solved Factorisation Problems
Here are a few more step-by-step factorisation problems for better understanding:
1. Factorise \( y^2 + 16y + 60 \):- Split: \( y^2 + 10y + 6y + 60 \)
- Group: \( (y^2 + 10y) + (6y + 60) \)
- Factor: \( y(y + 10) + 6(y + 10) = (y + 6)(y + 10) \)
2. Factorise \( 5x^2 + 14x – 3 \):
- Split: \( 5x^2 – x + 15x – 3 \)
- Group: \( (5x^2 – x) + (15x – 3) \)
- Factor: \( x(5x – 1) + 3(5x – 1) = (x + 3)(5x – 1) \)
3. Factorise \( 6a^2b – 8ab + 10ab^2 \):
- Take out common factor: \( 2ab(3a – 4 + 5b) \)
For more examples, visit Factorisation (Concept Overview) or download PDFs for self-practice.
Practice Problems
- Factorise \( x^2 + 7x + 10 \).
- Find the factors of \( 9y^2 – 25 \).
- Factorise \( a^2 – 2ab + b^2 \).
- Break down \( 12pq – 16q^2 + 20q \) into its factors.
- Challenge: Factorise \( x^4 – y^4 \).
Common Mistakes to Avoid
- Forgetting to take out the highest common factor at the start.
- Mixing up algebraic factorisation with prime factorisation of numbers.
- Skipping the verification step by multiplying your factors back.
- Rushing and missing suitable algebraic identities.
- Leaving expressions partially factored.
Real-World Applications
The concept of factorisation problems appears in areas such as physics equations, engineering designs, statistics, and finance. Knowing how to factorise efficiently helps not just in exams but also in understanding calculations in everyday life. Vedantu helps students see how maths applies beyond the classroom with interactive lessons and practical worksheets.
Download Factorisation Problems PDF
For more practice on factorisation problems, download comprehensive worksheets and solutions from this PDF resource.
Linked Topics and Further Reading
- Quadratic Equations: Learn how factorisation solves quadratic equations easily.
- Factoring Polynomials: Master advanced polynomial factorisation techniques.
- Algebraic Expressions: Build foundations for factorisation with expression structure.
- Prime Factorization: Know the difference between prime and algebraic factorisation.
- Polynomial: Dive deeper into polynomial properties and operations.
- Maths Formulas for Class 8: Access all key factorisation formulas for class 8 and 9 student exams.
- Factorization of Algebraic Expressions: Practice higher-level examples for Olympiads and board prep.
- HCF by Long Division Method: Strengthen your concept of common factors and applications.
We explored the idea of factorisation problems, how to apply different methods, solve related problems step by step, and understand their real-life relevance. Practice more with Vedantu to build confidence in these concepts and excel in your maths exams.
FAQs on Factorisation Problems in Algebra Explained
1. What is factorisation in Maths?
Factorisation is the process of writing an expression as a product of two or more simpler expressions called factors. In algebra, it means breaking down a polynomial into multiplying parts.
- Example: x² − 9 = (x − 3)(x + 3)
- Example: 6x = 2 × 3 × x
2. How do you factorise a quadratic expression?
To factorise a quadratic expression ax² + bx + c, find two numbers that multiply to ac and add to b. Follow these steps:
- Multiply a × c
- Find two numbers that multiply to ac and add to b
- Split the middle term
- Factor by grouping
- Numbers that multiply to 6 and add to 5 are 2 and 3
- x² + 5x + 6 = (x + 2)(x + 3)
3. What is the difference between factorisation and expansion?
Factorisation breaks an expression into brackets, while expansion multiplies brackets to form a single expression. They are opposite processes.
- Factorisation: x² − 4 = (x − 2)(x + 2)
- Expansion: (x − 2)(x + 2) = x² − 4
4. How do you factorise by taking out a common factor?
To factorise by taking out a common factor, identify the greatest common factor (GCF) of all terms and divide each term by it. Steps:
- Find the highest common factor (HCF)
- Divide each term by the HCF
- Write the expression as HCF × bracket
- HCF = 3x
- 6x² + 9x = 3x(2x + 3)
5. What is factorising by grouping?
Factorising by grouping is a method used for four-term expressions where terms are grouped in pairs to find common factors. Steps:
- Group terms in pairs
- Factor out common factors from each pair
- Factor out the common bracket
- (x² + 3x) + (2x + 6)
- x(x + 3) + 2(x + 3)
- (x + 3)(x + 2)
6. How do you factorise the difference of two squares?
The difference of two squares follows the formula a² − b² = (a − b)(a + b). This applies when both terms are perfect squares and separated by subtraction.
- Example: x² − 16
- Since 16 = 4²
- x² − 16 = (x − 4)(x + 4)
7. Can you factorise expressions with a leading coefficient greater than 1?
Yes, quadratics with a leading coefficient greater than 1 can be factorised using the ac method or splitting the middle term. Example: 2x² + 7x + 3
- Multiply 2 × 3 = 6
- Numbers that multiply to 6 and add to 7 are 6 and 1
- Rewrite: 2x² + 6x + x + 3
- Factor: (2x + 1)(x + 3)
8. Why is factorisation important in solving equations?
Factorisation is important because it allows us to solve equations using the zero product property. If a × b = 0, then a = 0 or b = 0.
- Example: x² − 5x + 6 = 0
- Factor: (x − 2)(x − 3) = 0
- Solutions: x = 2 or x = 3
9. What are common mistakes in factorisation?
Common mistakes in factorisation include sign errors, missing common factors, and incorrect number pairs. Watch out for:
- Forgetting to factor out the greatest common factor first
- Choosing numbers that multiply correctly but do not add to the middle term
- Sign mistakes in expressions like x² − 5x + 6
10. What is the formula for factorising a quadratic equation?
There is no single formula for factorising, but the related solving formula is the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. If the discriminant b² − 4ac is a perfect square, the quadratic can usually be factorised easily.
- Example: For x² − 5x + 6
- b² − 4ac = 25 − 24 = 1
- Since 1 is a perfect square, it factorises as (x − 2)(x − 3)
