Factorisation — Lesson
1) Hook — A Fun Real-Life Story to Grab Attention
Imagine you are helping your friend arrange gift boxes for Diwali. Each box contains sweets arranged in rows and columns. To pack them efficiently, you want to find the best way to group the sweets without leftovers. This is exactly what factorisation helps us do in algebra — breaking down expressions into simpler parts to understand and solve problems easily!
2) Core Concepts — Clear Explanation with Examples
Factorisation means expressing a polynomial as a product of its factors (simpler polynomials or numbers).
Why factorise?
- To simplify expressions
- To solve quadratic equations easily
- To find common factors in algebraic expressions
Common methods of factorisation:
| Method | Explanation | Example |
|---|---|---|
| Taking Common Factor | Identify and take out the greatest common factor (GCF) from all terms. | 12x + 18 = 6(2x + 3) |
| Factorisation of Quadratic Trinomials | Express ax² + bx + c as (mx + n)(px + q) | x² + 5x + 6 = (x + 2)(x + 3) |
| Difference of Squares | a² - b² = (a - b)(a + b) | x² - 16 = (x - 4)(x + 4) |
| Perfect Square Trinomial | a² ± 2ab + b² = (a ± b)² | x² + 6x + 9 = (x + 3)² |
Example 1: Taking Common Factor
Factorise: 15xy + 25x²
Solution: GCF is 5x
Answer: 5x(3y + 5x)
Example 2: Factorise x² + 7x + 12
Find two numbers whose product = 12 and sum = 7 → 3 and 4
Answer: (x + 3)(x + 4)
3) Key Formulas / Rules
1. Taking Common Factor: a·x + a·y = a(x + y)
2. Difference of Squares: a² - b² = (a - b)(a + b)
3. Perfect Square Trinomial: a² + 2ab + b² = (a + b)²
4. Quadratic Trinomial: x² + (m + n)x + mn = (x + m)(x + n)
4) Did You Know?
Factorisation is one of the oldest mathematical techniques, used in ancient India by mathematicians like Bhaskara II to solve equations and understand numbers. The word "factor" itself comes from the Latin word facere meaning "to make" or "to do".
5) Exam Tips — Common Mistakes & Board Exam Patterns
- Always look for the greatest common factor first. Missing this step wastes time and leads to incorrect answers.
- Check signs carefully when factorising quadratics — positive and negative signs change factors completely.
- Practice difference of squares and perfect square trinomials as they frequently appear in ICSE board exams.
- Remember the mnemonic "GCF, DOTS, PST, QT" to recall methods: Greatest Common Factor, Difference Of Two Squares, Perfect Square Trinomial, Quadratic Trinomial.
- Board exam pattern: Factorisation questions often come as 2-3 marks each and may involve multiple methods in one question.
- Always verify your factorisation by expanding the factors to get the original expression.
Factorisation — Mcq
Factorisation — Mnemonic
Mnemonic 1: "F.A.C.T.O.R" for Remembering Steps of Factorisation 🧮✨
- F - Find the Greatest Common Factor (GCF) first 🔍
- A - Apply special identities (like a² - b²) 📐
- C - Check for Common Binomial Factors 🔄
- T - Try splitting the middle term (ax² + bx + c) ✂️
- O - Organize terms to group and factor by grouping 🤝
- R - Rewrite fully factored form ✅
Mnemonic phrase: "Find All Clever Tricks Of Roots"
Mnemonic 2: Hindi Rhyming Trick for Special Products 📏🎶
"A² - B² = (A - B)(A + B) yaad rakhne ka asaan tareeka:
“Aapne Apna Bina Baat Ke Bichoda”
(Aapne Apna = A², Bina Baat = B², Ke Bichoda = (A - B)(A + B))
Isse yaad rahega ki difference of squares ka factorisation hota hai (A - B)(A + B)!
Mnemonic 3: Funny Acronym for Splitting the Middle Term ✂️➗
"M.I.D.D.L.E" helps to remember middle term splitting:
- M - Multiply first and last coefficients (a × c)
- I - Identify two numbers whose product = a×c and sum = b
- D - Divide the middle term into two terms using those numbers
- D - Do grouping of terms in pairs
- L - Look for common factors in each group
- E - Extract common binomial factor
Remember: "My Intelligent Dog Does Lots of Exercises" 🐶📚
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