AS Mathematics Algebra – Manipulation of brackets

Objectives Be confident in the use of brackets Be able to factorise linear expressions

Review of expanding brackets Example 1 Expand (x + 3)(x + 5) x2x2 + 8x+ 15 x2x2 + 5x+ 3x+ 15 multiply term by term collect like terms

Alternatives for expanding (x + 3)(x + 5) smiley face grid method (x+3)(x+5) x+5 x +3 x2x2 +5x +3x+15

Example 2 Expand(x + 4) 2 x2x2 + 8x+ 16 x2x2 + 4x + 16 (x + 4) Perfect square! Remember(a + b) 2 = a 2 + 2ab + b 2

Example 3 x2x x2x2 + 8x- 8x- 64 Expand(x - 8)(x + 8) Difference of 2 squares! Remember(a-b)(a+b) = a 2 - b 2

+ x - 2 Example 4 x 3 - 2x 2 Expand (x 2 + 2x + 1)(x - 2) multiply term by term – A harder one! x3x3 -3x x 2 – 4x collect like terms

Expand (x + 4)(x - 3)(2x + 1) Example 5 (x 2 + x – 12)(2x + 1) 2x 3 + x 2 + 2x 2 + x – 24x x 3 + 3x 2 – 23x - 12 multiply any two brackets collect like terms multiply remaining bracket

Factorising This involves taking out any common factors. Try to spot the HCF by inspection.

(i) common factors Example 1 Factorise 12x – 18y 6(2x – 3y) Example 2 Factorise 6x 2 – 21x 3x(2x – 7) The HCF of 12x & 18y is 6 The HCF of 6x 2 & 21x is 3x Check your answer by expanding the brackets

(ii) grouping Example 3 Factorise ax + ay + bx + by a(x + y) + b(x + y) (x + y)(a + b) The first two terms have common factor a, the last two have common factor y There is now a common factor of (x + y) Check your answer by expanding the brackets.

To illustrate this:- az + bz a(x + y) + b(x + y) z(a + b) …..as before! let z = x + y (x + y)(a + b) but z = x + y

Example 4 Factorise xy + 2x + 3y + 6 x(y + 2) + 3(y + 2) (y + 2)(x + 3) The first two terms have common factor x, the last two have common factor 3 There is now a common factor of (y + 2) Check your answer by expanding the brackets.

Example 5 Factorise 2x - 2xy - y + y 2 2x(1 - y) - y(1 - y) (1 - y)(2x - y) Check your answer by expanding the brackets For this method to succeed, both brackets should be the same, i.e both (1 - y) Example 6 Factorise 6a + 3ab - 2b - b 2 3a(2 + b) - b(2 + b) (2 + b)(3a - b)