1. AS Mathematics Algebra – Manipulation of brackets

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

3. 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

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

5. 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

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

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

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

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

10. (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

11. (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.

12. 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

13. 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.

14. 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)