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Chapter Analysis
Intermediate15 pages • EnglishQuick Summary
In this chapter, students explore the process of factorisation, which is the method of expressing an algebraic expression as a product of its factors. The chapter covers various techniques to factorise expressions using identities, common factors, and regrouping. It also delves into the division of algebraic expressions, highlighting the systematic approach to factorising trinomial and other complex expressions. Key concepts like irreducible factors and the role of identities in simplifying expressions are emphasized.
Key Topics
- •Factorisation using common factors
- •Factorisation using identities
- •Factorisation by regrouping terms
- •Division of algebraic expressions
- •Difference of squares
- •Factorisation of trinomials
- •Identifying irreducible factors
- •Application of distributive law
Learning Objectives
- ✓Understand the concept of factorisation and its importance in algebra
- ✓Develop proficiency in factorising expressions using common factors
- ✓Learn to use algebraic identities to simplify and factorise expressions
- ✓Master the division of polynomial expressions by monomials and other polynomials
- ✓Identify and extract irreducible factors in algebraic expressions
- ✓Apply systematic techniques to factorise trinomials and other complex expressions
Questions in Chapter
Find the common factors of the given terms: (i) 12x, 36 (ii) 2y, 22xy (iii) 14 pq, 28p2q2 (iv) 2x, 3x2, 4 (v) 6 abc, 24ab2, 12 a2b (vi) 16 x3, - 4x2, 32x (vii) 10 pq, 20qr, 30rp (viii) 3x2 y3, 10x3 y2,6 x2 y2z
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Factorise the following expressions: (i) 7x - 42 (ii) 6p - 12q (iii) 7a2 + 14a (iv) - 16 z + 20 z3 (v) 20 l2 m + 30 a l m (vi) 5 x2 y - 15 xy2 (vii) 10 a2 - 15 b2 + 20 c2 (viii) - 4 a2 + 4 ab - 4 ca (ix) x2 y z + x y2z + x y z2 (x) a x2 y + b x y2 + c x y z
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Additional Practice Questions
Factorise the expression 2x + 6.
easyAnswer: 2x + 6 can be written as 2(x + 3). The terms have a common factor of 2.
If x2 + 7x + 10 is factorised into (x + a)(x + b), find a and b.
mediumAnswer: The expression x2 + 7x + 10 can be factorised by finding two numbers that multiply to 10 and add up to 7. These numbers are 5 and 2. Therefore, the factorisation is (x + 5)(x + 2).
Factorise the quadratic expression y2 - 9.
mediumAnswer: The expression y2 - 9 is a difference of two squares and can be factorised as (y - 3)(y + 3).
Given that m2 - 4m - 12 = (m + p)(m + q), determine p and q.
mediumAnswer: To factorise m2 - 4m - 12, find two numbers whose product is -12 and sum is -4. The numbers are -6 and 2. Therefore, the factorisation is (m - 6)(m + 2).
Simplify and factorise the expression 4xy - 6x + 8y - 12.
hardAnswer: Group the terms: (4xy - 6x) + (8y - 12) gives x(4y - 6) + 4(2y - 3). Factor further to obtain the expression (4y - 6)(x + 2).