Cambridge International Mathematics

36 Algebra (Expansion and factorisation) (Chapter 1)

Example 8 Self Tutor

Expand and simplify:

a (2x¡3)(2x+3) b (5¡ 3 y)(5 + 3y) c (3x+4y)(3x¡ 4 y)

a (2x¡3)(2x+3)

=(2x)^2 ¡ 32

=4x^2 ¡ 9

b (5¡ 3 y)(5 + 3y)

=5^2 ¡(3y)^2

=25¡ 9 y^2

c (3x+4y)(3x¡ 4 y)

=(3x)^2 ¡(4y)^2

=9x^2 ¡ 16 y^2

EXERCISE 1C

1 Expand and simplify using the rule (a+b)(a¡b)=a^2 ¡b^2 :

a (x+ 2)(x¡2) b (x¡2)(x+2) c (2 +x)(2¡x)

d (2¡x)(2 +x) e (x+ 1)(x¡1) f (1¡x)(1 +x)

g (x+ 7)(x¡7) h (c+ 8)(c¡8) i (d¡5)(d+5)

j (x+y)(x¡y) k (4 +d)(4¡d) l (5 +e)(5¡e)

2 Expand and simplify using the rule (a+b)(a¡b)=a^2 ¡b^2 :

a (2x¡1)(2x+1) b (3x+ 2)(3x¡2) c (4y¡5)(4y+5)

d (2y+ 5)(2y¡5) e (3x+ 1)(3x¡1) f (1¡ 3 x)(1 + 3x)

g (2¡ 5 y)(2 + 5y) h (3 + 4a)(3¡ 4 a) i (4 + 3a)(4¡ 3 a)

3 Expand and simplify using the rule (a+b)(a¡b)=a^2 ¡b^2 :

a (2a+b)(2a¡b) b (a¡ 2 b)(a+2b) c (4x+y)(4x¡y)

d (4x+5y)(4x¡ 5 y) e (2x+3y)(2x¡ 3 y) f (7x¡ 2 y)(7x+2y)

4aUse the difference of two squares expansion to show that:

i 43 £37 = 40^2 ¡ 32 ii 24 £26 = 25^2 ¡ 12

b Evaluate without using a calculator:

i 18 £ 22 ii 49 £ 51 iii 103 £ 97 :

Discovery 1 The product of three consecutive integers

Con was trying to multiply 19 £ 20 £ 21 without a calculator. Aimee told him to ‘cube the middle

integer and then subtract the middle integer’ to get the answer.

What to do:

1 Find 19 £ 20 £ 21 using a calculator.

2 Find 203 ¡ 20 using a calculator. Does Aimee’s rule seem to work?

3 Check that Aimee’s rule works for the following products:

a 4 £ 5 £ 6 b 9 £ 10 £ 11 c 49 £ 50 £ 51

4 Let the middle integer bex, so the other integers must be(x¡1)and(x+1).

Find the product (x¡1)£x£(x+1) by expanding and simplifying. Have you proved Aimee’s

rule?

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Y:\HAESE\IGCSE01\IG01_01\036IGCSE01_01.CDR Wednesday, 10 September 2008 2:06:44 PM PETER