Algebra is the language of mathematics. In Class 8, we extend our algebraic skills to include algebraic identities, factorisation, and division of algebraic expressions. These tools are essential for solving equations and simplifying complex expressions.
Algebraic Identities
An identity is an equation that is true for all values of the variable(s).
- 1.Standard identities you must know:
- 2.(a + b)2 = a2 + 2ab + b2
- 3.(a - b)2 = a2 - 2ab + b2
- 4.(a + b)(a - b) = a2 - b2
- 5.(x + a)(x + b) = x2 + (a + b)x + ab
- 6.(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Factorisation
Factorisation means expressing an expression as a product of simpler expressions.
- Methods:
- Common factor: take out the HCF. e.g., 3x2 + 6x = 3x(x + 2)
- Grouping: group terms and factorise. e.g., ax + bx + ay + by = x(a+b) + y(a+b) = (x+y)(a+b)
- Using identities: recognise the pattern and apply identity.
- Factorising trinomials: x2 + (a+b)x + ab = (x+a)(x+b)
Division of Algebraic Expressions
Divide by cancelling common factors, similar to dividing fractions.
e.g., (6x3 + 3x2) / (3x) = 2x2 + x.
Expand (2x + 3y)2.
Using identity 1: a = 2x, b = 3y.
= (2x)2 + 2(2x)(3y) + (3y)2 = 4x2 + 12xy + 9y2.
Find the value of (103)2 using identity.
(103)2 = (100 + 3)2 = 1002 + 2×100×3 + 32 = 10000 + 600 + 9 = 10609.
Factorise x2 - 49.
Using identity 3: x2 - 49 = x2 - 72 = (x + 7)(x - 7).
Factorise 4x2 + 12x + 9.
= (2x)2 + 2(2x)(3) + 32 = (2x + 3)2.
Factorise x2 + 5x + 6.
Find two numbers with sum = 5 and product = 6: 2 and 3.
= (x + 2)(x + 3).
Divide (12x3 - 6x2) by 3x.
= (12x3)/(3x) - (6x2)/(3x) = 4x2 - 2x.
Factorise 2x + 4y + ax + 2ay by grouping.
= 2(x + 2y) + a(x + 2y) = (2 + a)(x + 2y).
- Key formulas:
- (a + b)2 = a2 + 2ab + b2
- (a - b)2 = a2 - 2ab + b2
- (a + b)(a - b) = a2 - b2
- (x + a)(x + b) = x2 + (a+b)x + ab
- Always check factorisation by expanding back.
Common mistakes
- Forgetting the middle term 2ab in (a±b)2.
- Incorrectly treating (a + b)2 as a2 + b2 (missing 2ab).
- Sign errors in (a - b)2 = a2 - 2ab + b2 (not + 2ab).
- In factorisation by grouping, choosing the wrong pairs.
Summary
The algebraic identities are shortcuts for expansion and factorisation. Factorisation is the reverse of expansion. These skills form the foundation for solving quadratic equations and simplifying rational expressions in higher classes.