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2. Refresher Brackets and Factorising In earlier work on brackets and factorising we completed the problems shown below and noted that factorising was simply the opposite of expanding brackets. (a) 4(x + 2)= 4x + 8 = 6x + 27 = 15x - 18 = 40 + 16x = 20p - 25 = 8x + 12y - 20 (b) 3(2x + 9) (c) 3(5x - 6) (d) 8(2x + 5) (e) 5(4p - 5) (f) 4(2x + 3y - 5) Expand Factorise In this earlier work the term outside the bracket was always a number.

3. Harder Brackets Brackets and Factorising + x + = + + x - = - - x + = - - x - = +- x - = + Harder Brackets: The term outside a bracket is not always a number, it may contain letters and numbers. Example 1 Expand x(x + 9) = x 2 + 9x Example 2 Expand 2x(3x - p) = 6x 2 - 2xp Example 3 Expand 4a(2ax - 1) = 8a 2 x - 4a Example 4 Expand x(2x 2 y - m) = 2x 3 y - xm Remember: 2 ar = 2 x a x r ab x cd = abcd 2 x x 3 x = 6x 2 xy x 2y = 2xy 2 Also: x x x x x = x 3 x x x 2 = x 3 2 x 2 x 3 x = 6x 3 xy 2 x yx = x 2 y 3

4. Questions 1 Brackets and Factorising Expand each of the following: = x 2 + 8x = x 3 - 4x = 6ax - 4a = 10a 2 b 2 + 5ab = p 3 q - 4pq = 2a 3 x 3 - 2a 2 x 2 (a) x(x + 8) (b) x(x 2 - 4) (c) 2a(3x - 2) (d) 5ab(2ab + 1) (e) pq(p 2 - 4) (f) 2ax 2 (a 2 x - a)

5. Harder Factorising Brackets and Factorising + x + = + + x - = - - x + = - - x - = +- x - = + Harder Factorisation: Example 1 Factorise x 2 + 7x = x(x + 7) Example 2 Factorise 6x 3 - 3x 2 = 3x 2 (2x - 1) Example 3 Factorise 8a 2 x -12a = 4a(2ax - 3) Example 4 Factorise 2x 3 y - x 2 y = x 2 y(2x - 1) Remember: 2 ar = 2 x a x r ab x cd = abcd 2 x x 3 x = 6x 2 xy x 2y = 2xy 2 Also: x x x x x = x 3 x x x 2 = x 3 2 x 2 x 3 x = 6x 3 xy 2 x yx = x 2 y 3

6. Brackets and Factorising (a) x 2 + 6x (b) x 3 - 7x (c) 9ax - 12a (d) 6a 2 b 2 + 3ab (e) p 3 q - 8pq (f) 3a 3 x 3 - 3a 2 x 2 = x(x + 6) = x(x 2 - 7) = 3a(3x - 4) = 3ab(2ab + 1) = pq(p 2 - 8) = 3a 2 x 2 (ax - 1) Factorise each of the following: Questions 2

7. Double Brackets 1 Expansion of Double Brackets (x + 5)(x + 6) Expanding double brackets can be reduced to expanding two single brackets. This expression can be thought of as x + 5 lots of x + 6. So we can simply work out x lots of x + 6 first then add on 5 lots of x + 6. (x + 5)(x + 6) = = x 2 + 6x + 5x + 30 = x 2 + 11x + 30 x5 + x 6 + x2x2 5x5x 6x6x30 + x(x + 6) 5(x + 6) It is sometimes useful to think of a problem like this geometrically.

8. Expansion of Double Brackets + x + = + + x - = - - x + = - - x - = +- x - = + (x + 2)(x + 3) Expand Example Question 1 = x(x + 3) + 2(x + 3) = x 2 + 3x + 2x + 6 = x 2 + 5x + 6 (x + 4)(x - 7) Expand Example Question 2 = x(x - 7) + 4(x - 7) = x 2 - 7x + 4x - 28 = x 2 - 3x - 28 (x - 3) 2 Expand Example Question 3 = x(x - 3) - 3(x - 3) = x 2 - 3x - 3x + 9 = x 2 - 6x + 9 = (x - 3)(x - 3)

9. Questions 3 Expansion of Double Brackets + x + = + + x - = - - x + = - - x - = +- x - = + x 2 + 9x + 20 x 2 + 10x + 21 x 2 + 2x + 1 x 2 + x - 12 x 2 - 7x + 12 x 2 - 2x - 24 Expand each of the following: (a) (x + 5)(x + 4) (b) (x + 7)(x + 3) (c) (x + 1) 2 (d) (x + 4)(x - 3) (e) (x - 4)(x - 3) (f) (x - 6)(x + 4) (g) (x + 5)(x - 5) x 2 - 25

10. Expansion of Double Brackets + x + = + + x - = - - x + = - - x - = +- x - = + (2x + 1)(x + 3) Expand Example Question 4 = 2x(x + 3) + 1(x + 3) = 2x 2 + 6x + x + 3 = 2x 2 + 7x + 3 (x + 4)(3x - 7) Expand Example Question 5 = x(3x - 7) + 4(3x - 7) = 3x 2 - 7x + 12x - 28 = 3x 2 + 5x - 28 (2x - 3)(5x - 4) Expand Example Question 6 = 2x(5x - 4) - 3(5x - 4) = 10x 2 - 8x - 15x + 12 = 10x 2 - 23x + 12

11. Questions 4 Expansion of Double Brackets + x + = + + x - = - - x + = - - x - = +- x - = + (a) 2x 2 + 13x + 20 (b) 4x 2 + 31x + 21 (c) 2x 2 + 5x + 3 (d) 12x 2 + 7x - 12 (e) 12x 2 - 25x + 12 (f) 10x 2 - 17x - 20 Expand each of the following: (a) (2x + 5)(x + 4) (b) (x + 7)(4x + 3) (c) (x + 1)(2x + 3) (d) (3x + 4)(4x - 3) (e) (3x - 4)(4x - 3) (f) (2x - 5)(5x + 4) (g) (3x + 5)(3x - 5) (g) 9x 2 - 25

12. DB 2 FOIL Expansion of Double Brackets An alternative way to expand double brackets without having to write out the two single brackets is as follows. (x + 6)(x + 5) = = x 2 + 11x + 30 F irst O uter I nner L ast x2x2 + 5x+ 6x+ 30 (2x + 4)(x - 3) = = 2x 2 - 2x + 30 F irst O uter I nner L ast 2x22x2 - 6x + 4x - 12 Some people like to use the mnemonic FOIL to help them remember.

13. Expansion of Double Brackets = x 2 + 2x + 4x + 8 Expand the following using this method: = x 2 - 8x + 3x - 24 = 3x 2 - 12x - x + 4 = 6x 2 - 6x - 4x + 4 (x + 4)(x + 2) (x + 3)(x - 8) (3x - 1)(x - 4) (3x - 2)(2x - 2) 1 2 3 4 = x 2 + 6x + 8 = x 2 - 5x - 24 = 3x 2 - 13x + 4 = 6x 2 - 10x + 4

14. Questions 5 Expansion of Double Brackets + x + = + + x - = - - x + = - - x - = +- x - = + = x 2 + 9x + 14 = x 2 + 11x + 24 = x 2 - 3x - 4 = x 2 + 2x - 15 = x 2 - 8x + 15 = 2y 2 + 2y - 24 Expand each of the following using the method outlined. (a) (x + 2)(x + 7) (b) (x + 8)(x + 3) (c) (x + 1)(x - 4) (d) (x + 5)(x - 3) (e) (x - 5)(x - 3) (f) (2y - 6)(y + 4) (g) (5x + 6)(2x - 6) = 10x 2 - 18x - 36

15. Double Brackets 3 Expansion of Double Brackets It is a good idea to learn how to expand double brackets by inspection. It is quicker and more efficient than other methods as well as being useful in more advanced work. We will look at the example below to see how the method works. (x + 6)(x + 5) = x 2 + 11x + 30 The colours show how the x 2 term and the constant term are formed by multiplication and how easy it is to simply “write down” their values instantly. The remaining middle term in x is obtained as follows: Simply multiply the inner terms and outer terms and then add them. 11x (x + 4)(x + 3) Use the method to expand the following. = x 2 + 7x + 12

16. Expansion of Double Brackets Care needs to be taking when negative signs are involved particularly when obtaining the middle term. (x + 4)(x - 3) = x 2 + x - 12 4 x -3 4 x - 3 x (x - 4)(x - 3) = x 2 - 7x + 12 -4 x -3 -4 x - 3 x With practice you should be able to simply write the answer down in the normal order i.e. first term, second term and third term.

17. Expansion of Double Brackets Care needs to be taking when negative signs are involved particularly when obtaining the middle term. = x 2 - x - 6 Try and expand the following using this method: = x 2 + 3x - 10 = x 2 - 5x + 4 = x 2 - 6x + 9 (x + 2)(x - 3) (x - 2)(x + 5) (x - 1)(x - 4) (x - 3)(x - 3) 1 2 3 4

18. Questions 6 Expansion of Double Brackets + x + = + + x - = - - x + = - - x - = +- x - = + = x 2 + 9x + 14 = x 2 + 11x + 24 = x 2 - 3x - 4 = x 2 + 2x - 15 = x 2 - 8x + 15 = y 2 - 2y - 24 Expand each of the following using the method outlined. (a) (x + 2)(x + 7) (b) (x + 8)(x + 3) (c) (x + 1)(x - 4) (d) (x + 5)(x - 3) (e) (x - 5)(x - 3) (f) (y - 6)(y + 4) (g) (x + 6)(x - 6) = x 2 - 36

19. Expansion of Double Brackets Additional care needs to be taken when the coefficient of the x term is greater than 1. (2x + 4)(x - 3) = 2x 2 - 2x - 12 4 x -3 4 x - 6 x (3x - 4)(5x - 3) = 15x 2 - 29x + 12 -4 x -3 -20 x - 9 x 2 x X x 3 x X 5 x

20. Expansion of Double Brackets Care needs to be taken when negative signs are involved particularly when obtaining the middle term. = 2x 2 - 4x - 6 Expand the following by inspection: = 3x 2 + 13x - 10 = 6x 2 - 11x + 4 = 25x 2 - 30x + 9 (2x + 2)(x - 3) (3x - 2)(x + 5) (2x - 1)(3x - 4) (5x - 3)(5x - 3) 1 2 3 4

21. Questions 7 Expansion of Double Brackets + x + = + + x - = - - x + = - - x - = +- x - = + = 2x 2 + 16x + 14 = 3x 2 + 27x + 24 = 4x 2 - 15x - 4 = 6x 2 + x - 15 = 3x 2 - 14x + 15 = 20y 2 - 4y - 24 Expand each of the following using the method outlined. (a) (2x + 2)(x + 7) (b) (x + 8)(3x + 3) (c) (4x + 1)(x - 4) (d) (3x + 5)(2x - 3) (e) (3x - 5)(x - 3) (f) (5y - 6)(4y + 4) (g) (3x + 6)(3x - 6) = 9x 2 - 36

22. See Quadratics 1 Factorising = 2x 2 + 16x + 14 = 3x 2 + 27x + 24 = 4x 2 - 15x - 4 = 6x 2 + x - 15 = 3x 2 - 14x + 15 = 20y 2 - 4y - 24 (a) (2x + 2)(x + 7) (b) (x + 8)(3x + 3) (c) (4x + 1)(x - 4) (d) (3x + 5)(2x - 3) (e) (3x - 5)(x - 3) (f) (5y - 6)(4y + 4) (g) (3x + 6)(3x - 6)= 9x 2 - 36 Expand Factorise

23. Worksheet 1 (a) x(x + 8) (b) x(x 2 - 4) (c) 2a(3x - 2) (d) 5ab(2ab + 1) (e) pq(p 2 - 4) (f) 2ax 2 (a 2 x - a) 1 (a) x 2 + 6x (b) x 3 - 7x (c) 9ax - 12a (d) 6a 2 b 2 + 3ab (e) p 3 q - 8pq (f) 3a 3 x 3 - 3a 2 x 2 2 Expand Simplify (a) x(x + 8) (b) x(x 2 - 4) (c) 2a(3x - 2) (d) 5ab(2ab + 1) (e) pq(p 2 - 4) (f) 2ax 2 (a 2 x - a) 1 (a) x 2 + 6x (b) x 3 - 7x (c) 9ax - 12a (d) 6a 2 b 2 + 3ab (e) p 3 q - 8pq (f) 3a 3 x 3 - 3a 2 x 2 2 Expand Simplify

24. (a) (2x + 5)(x + 4) (b) (x + 7)(4x + 3) (c) (x + 1)(2x + 3) (d) (3x + 4)(4x - 3) (e) (3x - 4)(4x - 3) (f) (2x - 5)(5x + 4) (g) (3x + 5)(3x - 5) 4 (a) (x + 5)(x + 4) (b) (x + 7)(x + 3) (c) (x + 1) 2 (d) (x + 4)(x - 3) (e) (x - 4)(x - 3) (f) (x - 6)(x + 4) (g) (x + 5)(x - 5) 3 Worksheet 2 Expand using single brackets. Expand by inspection. (a) (x + 2)(x + 7) (b) (x + 8)(x + 3) (c) (x + 1)(x - 4) (d) (x + 5)(x - 3) (e) (x - 5)(x - 3) (f) (2y - 6)(y + 4) (g) (5x + 6)(2x - 6) 5 (a) (2x + 2)(x + 7) (b) (x + 8)(3x + 3) (c) (4x + 1)(x - 4) (d) (3x + 5)(2x - 3) (e) (3x - 5)(x - 3) (f) (5y - 6)(4y + 4) (g) (3x + 6)(3x - 6) 6 FOIL (a) (x + 2)(x + 7) (b) (x + 8)(x + 3) (c) (x + 1)(x - 4) (d) (x + 5)(x - 3) (e) (x - 5)(x - 3) (f) (y - 6)(y + 4) (g) (x + 6)(x - 6) 7