How to Solve Factorisation Problems Step by Step
The concept of factorisation in Maths plays a key role in algebra and is widely applicable to both real-life situations and exam scenarios. Learning how to break numbers and algebraic expressions into simpler factors makes calculations, problem-solving, and solving equations easier.
What Is Factorisation in Maths?
Factorisation in Maths is the process of breaking down a number or algebraic expression into a product of simpler numbers or expressions called factors. This concept is applied in areas such as prime factorisation of numbers, factorisation of algebraic expressions, and solving quadratic equations.
Why Do We Use Factorisation?
Factorisation is used to simplify complex expressions, solve equations, reduce fractions, and find solutions in algebra. It is essential for topics like Factoring Polynomials and Quadratic Equations. Proper factorisation helps you solve many examination and real-life problems efficiently.
Key Formulas for Factorisation in Maths
Here are some standard factorisation formulas you will use often:
- \( a^2 - b^2 = (a - b)(a + b) \)
- \( (a + b)^2 = a^2 + 2ab + b^2 \)
- \( (a - b)^2 = a^2 - 2ab + b^2 \)
- \( a^2 + b^2 = (a + b)^2 - 2ab \)
- \( a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2) \)
Remembering these formulas will help you factorise quickly during exams and while practicing sums at home.
Methods of Factorisation in Maths
There are a few major techniques for algebraic factorisation:
- Taking Common Factors: If terms have common numbers/variables, take them out as common factors.
- Regrouping or Splitting Terms: Rearranging the expression to reveal a common factor among grouped terms.
- Using Identities: Using algebraic identities to simplify and factorise (like a²–b², etc.).
- Splitting the Middle Term: Often used for quadratic expressions by splitting the coefficient of \(x\).
Each method is important for solving various kinds of factorisation questions in Maths. Practice will help you choose the best method for each case.
Step-by-Step Illustration
Let’s see how to factorise the expression \( x^2 + 8x + 15 \):
1. Look for two numbers whose sum is 8 and product is 15.2. The numbers are 3 and 5.
3. Rewrite the middle term: \( x^2 + 3x + 5x + 15 \)
4. Group and factor: \( x(x + 3) + 5(x + 3) = (x + 3)(x + 5) \)
5. Final Answer: \( (x + 3)(x + 5) \)
Speed Trick or Vedic Shortcut
Here’s a popular shortcut for factorising simple quadratic expressions:
- Write the quadratic in the form \( x^2 + p x + q \).
- Find two numbers whose sum is \( p \) and whose product is \( q \).
- Replace the middle term with two terms using those numbers, and factor by grouping.
Example: For \( x^2 + 7x + 10 \): 5 and 2 add to 7 and multiply to 10.
\( x^2 + 5x + 2x + 10 = x(x + 5) + 2(x + 5) = (x + 5)(x + 2) \).
Using such tricks can help you complete factorisation questions in seconds during competitive exams. More tips like this are taught in Vedantu’s interactive maths classes.
Try These Yourself
- Factorise: \( x^2 + 6x + 9 \)
- Factorise: \( 2x^2 + 7x + 3 \)
- Use the identity \( a^2 - b^2 \) to factorise \( 25y^2 - 16 \)
- Find the common factors in \( 4x^2y + 8xy^2 \)
Frequent Errors and Misunderstandings
- Not checking for a common factor in all terms first
- Forgetting or misusing algebraic identities
- Incorrect signs when splitting the middle term
- Confusing factorisation with simplification
Relation to Other Concepts
Mastering factorisation in Mathematics is essential before moving on to topics like Algebraic Identities, Algebraic Expressions, and Factoring Polynomials. It is also foundational for simplifying algebraic fractions and solving Polynomial Equations.
Classroom Tip
Teachers often use colour-coded steps and block diagrams to make the process of factorisation more visual. Try using coloured pens to highlight like terms or factors as you work through problems. At Vedantu, teachers use interactive screens to help students visualise each step of factorisation.
We explored factorisation in Maths—from definition, formula, stepwise examples, mistakes, and its connection with other algebra topics. Keep practicing factorisation sums to build your confidence and speed! For more tips, solved worksheets, and live coaching on tricky algebra problems, check out Vedantu’s classes and resources.
