2. Introduction This chapter focuses on basic manipulation of Algebra It also goes over rules of Surds and Indices It is essential that you understand this whole chapter as it links into most of the others!

4. Algebra and Functions Like Terms You can simplify expressions by collecting ‘like terms’ ‘Like Terms’ are terms that are the same, for example; 5x and 3x b 2 and -2b 2 7ab and 8ab are all ‘like terms’. 1A Examples a) b) c) Expand each bracket first

6. Algebra and Functions Indices (Powers) You need to be able to simplify expressions involving Indices, where appropriate. 1B

7. Algebra and Functions Indices (Powers) You need to be able to simplify expressions involving Indices, where appropriate. 1B Examples a) b) c) d) e) f)

9. Algebra and Functions Expanding Brackets You can ‘expand’ an expression by multiplying the terms inside the bracket by the term outside. 1C Examples a) b) c) d) e)

11. Algebra and Functions Factorising Factorising is the opposite of expanding brackets. An expression is put into brackets by looking for common factors. 1D a) Common Factor 3 b) x c) 4x d) 3xy e) 3x

13. Algebra and Functions Expand the following pairs of brackets (x + 4)(x + 7)  x 2 + 4x + 7x + 28  x 2 + 11x + 28 (x + 3)(x – 8)  x 2 + 3x – 8x – 24  x 2 – 5x - 24 + 28+ 7x+ 7 + 4xx2x2 x + 4x - 24- 8x- 8 + 3xx2x2 x + 3x

14. Algebra and Functions x2x2 +3x2+ You get the last number in a Quadratic Equation by multiplying the 2 numbers in the brackets You get the middle number by adding the 2 numbers in the brackets (x + 2)(x + 1)

15. Algebra and Functions x2x2 -2x15- You get the last number in a Quadratic Equation by multiplying the 2 numbers in the brackets You get the middle number by adding the 2 numbers in the brackets (x - 5)(x + 3)

16. Algebra and Functions x 2 - 7x + 12 Numbers that multiply to give + 12 +3 +4 -3 -4 +12 +1 -12 -1 +6 +2 -6 -2 Which pair adds to give -7? (x - 3)(x - 4) So the brackets were originally…

17. Algebra and Functions x 2 + 10x + 16 Numbers that multiply to give + 16 +1 +16 -1 -16 +2 +8 -2 -8 +4 -4 Which pair adds to give +10? (x + 2)(x + 8) So the brackets were originally…

18. Algebra and Functions x 2 - x - 20 Numbers that multiply to give - 20 +1 -20 -1 +20 +2 -10 -2 +10 +4 -5 -4 +5 Which pair adds to give - 1? (x + 4)(x - 5) So the brackets were originally…

19. Algebra and Functions Factorising Quadratics A Quadratic Equation has the form; ax 2 + bx + c Where a, b and c are constants and a ≠ 0. You can also Factorise these equations. REMEMBER  An equation with an ‘x 2 ’ in does not necessarily go into 2 brackets. You use 2 brackets when there are NO ‘Common Factors’ 1E Examples a) The 2 numbers in brackets must:  Multiply to give ‘c’  Add to give ‘b’

20. Algebra and Functions Factorising Quadratics A Quadratic Equation has the form; ax 2 + bx + c Where a, b and c are constants and a ≠ 0. You can also Factorise these equations. 1E Examples b) The 2 numbers in brackets must:  Multiply to give ‘c’  Add to give ‘b’

21. Algebra and Functions Factorising Quadratics A Quadratic Equation has the form; ax 2 + bx + c Where a, b and c are constants and a ≠ 0. You can also Factorise these equations. 1E Examples c) The 2 numbers in brackets must:  Multiply to give ‘c’  Add to give ‘b’ (In this case, b = 0) This is known as ‘the difference of two squares’  x 2 – y 2 = (x + y)(x – y)

22. Algebra and Functions Factorising Quadratics A Quadratic Equation has the form; ax 2 + bx + c Where a, b and c are constants and a ≠ 0. You can also Factorise these equations. 1E Examples d) The 2 numbers in brackets must:  Multiply to give ‘c’  Add to give ‘b’

23. Algebra and Functions Factorising Quadratics A Quadratic Equation has the form; ax 2 + bx + c Where a, b and c are constants and a ≠ 0. You can also Factorise these equations. 1E Examples d) The 2 numbers in brackets must:  Multiply to give ‘c’  Add to give ‘b’  Sometimes, you need to remove a ‘common factor’ first…

24. Algebra and Functions Expand the following pairs of brackets (x + 3)(x + 4)  x 2 + 3x + 4x + 12  x 2 + 7x + 12 (2x + 3)(x + 4)  2x 2 + 3x + 8x + 12  2x 2 + 11x + 12 + 12+ 4x+ 4 + 3xx2x2 x + 3x + 12+ 8x+ 4 + 3x2x 2 x + 32x When an x term has a ‘2’ coefficient, the rules are different… 2 of the terms are doubled  So, the numbers in the brackets add to give the x term, WHEN ONE HAS BEEN DOUBLED FIRST

25. Algebra and Functions 2x 2 - 5x - 3 Numbers that multiply to give - 3 -3 +1 +3 -1 One of the values to the left will be doubled when the brackets are expanded (2x + 1)(x - 3) So the brackets were originally… -6 +1 -3 +2 +6 -1 +3 -2 The -3 doubles so it must be on the opposite side to the ‘2x’

26. Algebra and Functions 2x 2 + 13x + 11 Numbers that multiply to give + 11 +11 +1 -11 -1 One of the values to the left will be doubled when the brackets are expanded (2x + 11)(x + 1) So the brackets were originally… +22 +1 +11 +2 -22 -1 -11 -2 The +1 doubles so it must be on the opposite side to the ‘2x’

27. Algebra and Functions 3x 2 - 11x - 4 Numbers that multiply to give - 4 +2 -2 -4 +1 +4 -1 One of the values to the left will be tripled when the brackets are expanded (3x + 1)(x - 4) So the brackets were originally… +6 -2 +2 -6 -12 +1 -4 +3 The -4 triples so it must be on the opposite side to the ‘3x’ +12 -1 +4 -3

29. Algebra and Functions Extending the rules of Indices The rules of indices can also be applied to rational numbers (numbers that can be written as a fraction) 1F Examples a) b) c) d)

30. Algebra and Functions Extending the rules of Indices The rules of indices can also be applied to rational numbers (numbers that can be written as a fraction) 1F Examples a) b) c) d)

31. Algebra and Functions Extending the rules of Indices The rules of indices can also be applied to rational numbers (numbers that can be written as a fraction) 1F Examples a) b)

33. Algebra and Functions Surd Manipulation You can use surds to represent exact values. 1G Examples Simplify the following… a) b) c)

35. Algebra and Functions Rationalising Rationalising is the process where a Surd is moved from the bottom of a fraction, to the top. 1H Multiply top and bottom by Examples Rationalise the following… a)

36. Algebra and Functions Rationalising Rationalising is the process where a Surd is moved from the bottom of a fraction, to the top. 1H Multiply top and bottom by Examples Rationalise the following… b)

37. Algebra and Functions Rationalising Rationalising is the process where a Surd is moved from the bottom of a fraction, to the top. 1H Multiply top and bottom by Examples Rationalise the following… c)

38. Summary We have recapped our knowledge of GCSE level maths We have looked at Indices, Brackets and Surds Ensure you master these as they link into the vast majority of A-level topics!