How to Solve Factorisation Using Common Factors and Identities
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.
FAQs on Factorisation of Algebraic Expressions and Polynomials
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 smaller algebraic parts that multiply together to give the original expression. For example:
- 6x = 2 × 3 × x
- x² − 9 = (x − 3)(x + 3)
2. How do you factorise a quadratic expression?
To factorise a quadratic expression of the form ax² + bx + c, split the middle term and group the terms. Steps:
- Multiply a × c
- Find two numbers that multiply to give ac and add to give b
- Split the middle term
- Factor by grouping
- Find numbers that multiply to 6 and add to 5 → 2 and 3
- x² + 2x + 3x + 6
- x(x + 2) + 3(x + 2)
- (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. In simple terms:
- Factorisation: x² + 5x + 6 → (x + 2)(x + 3)
- Expansion: (x + 2)(x + 3) → x² + 5x + 6
4. How do you factorise by taking out a common factor?
Factorising by taking out a common factor means dividing each term by the greatest common factor (GCF). Steps:
- Identify the greatest common factor
- Divide each term by it
- Write the result inside brackets
- GCF = 3x
- 3x(2x + 3)
5. What is the formula for the difference of two squares?
The difference of two squares formula is a² − b² = (a − b)(a + b). This identity works when both terms are perfect squares and separated by subtraction. Example:
- x² − 16
- 16 = 4²
- (x − 4)(x + 4)
6. How do you factorise trinomials?
To factorise a trinomial, write it as a product of two binomials whose product equals the original expression. For x² + bx + c:
- Find two numbers that multiply to c
- Add to give b
- Write as (x + m)(x + n)
- Numbers: 3 and 4
- (x + 3)(x + 4)
7. Why is factorisation important in solving equations?
Factorisation is important because it helps solve equations by using the zero product rule. If a × b = 0, then either a = 0 or b = 0. Example:
- x² − 5x + 6 = 0
- (x − 2)(x − 3) = 0
- x = 2 or x = 3
8. Can you give an example of factorising by grouping?
Factorising by grouping involves pairing terms to find common factors. Example: ax + ay + bx + by
- Group terms: (ax + ay) + (bx + by)
- Factor each group: a(x + y) + b(x + y)
- Factor common bracket: (a + b)(x + y)
9. What are common mistakes in factorisation?
Common mistakes in factorisation include incorrect sign handling and missing common factors. Typical errors:
- Forgetting to take out the greatest common factor first
- Mixing up positive and negative signs
- Incorrect multiplication check
10. How do you check if factorisation is correct?
You check factorisation by expanding the brackets and confirming you get the original expression. Steps:
- Multiply the factors using expansion
- Simplify the result
- Compare with the original polynomial
- (x + 2)(x + 3)
- Expand → x² + 5x + 6
- If it matches, the factorisation is correct.
