1. Algebra and Functions

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!

3. Teachings for Exercise 1A

4. Expand each bracket first
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
b2 and -2b2
7ab and 8ab
are all ‘like terms’.
Examples a) b) c)
Expand each bracket first 1A

5. Teachings for Exercise 1B

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

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

8. Teachings for Exercise 1C

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

10. Teachings for Exercise 1D

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

12. Teachings for Exercise 1E

13. Algebra and Functions Expand the following pairs of brackets
(x + 4)(x + 7)
 x2 + 4x + 7x + 28
 x2 + 11x + 28
(x + 3)(x – 8)
 x2 + 3x – 8x – 24
 x2 – 5x - 24 x
+ 4 x x2
+ 4x
+ 7
+ 7x
+ 28 x
+ 3 x x2
+ 3x
- 8
- 8x
- 24

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

15. Algebra and Functions (x - 5)(x + 3) x2 - 2x - 15
You get the middle number by adding the 2 numbers in the brackets
You get the last number in a Quadratic Equation by multiplying the 2 numbers in the brackets

16. Algebra and Functions x2 - 7x + 12 (x - 3)(x - 4)
Numbers that multiply to give + 12
Which pair adds to give -7?
+3 +4
-3 -4
+6 +2
-6 -2
So the brackets were originally…
(x - 3)(x - 4)

17. Algebra and Functions x2 + 10x + 16 (x + 2)(x + 8)
Numbers that multiply to give + 16
Which pair adds to give +10?
+2 +8
-2 -8
+4 +4
-4 -4
So the brackets were originally…
(x + 2)(x + 8)

18. Algebra and Functions x2 - x - 20 (x + 4)(x - 5)
Numbers that multiply to give - 20
Which pair adds to give - 1?
+4 -5
-4 +5
So the brackets were originally…
(x + 4)(x - 5)

19. Algebra and Functions 1E Factorising Quadratics
A Quadratic Equation has the form;
ax2 + bx + c
Where a, b and c are constants and a ≠ 0.
You can also Factorise these equations.
REMEMBER
 An equation with an ‘x2’ in does not necessarily go into 2 brackets. You use 2 brackets when there are NO ‘Common Factors’
Examples a)
The 2 numbers in brackets must:
Multiply to give ‘c’
Add to give ‘b’ 1E

20. Algebra and Functions 1E Factorising Quadratics
A Quadratic Equation has the form;
ax2 + bx + c
Where a, b and c are constants and a ≠ 0.
You can also Factorise these equations.
Examples b)
The 2 numbers in brackets must:
Multiply to give ‘c’
Add to give ‘b’ 1E

21. This is known as ‘the difference of two squares’
Algebra and Functions
Factorising Quadratics
A Quadratic Equation has the form;
ax2 + bx + c
Where a, b and c are constants and a ≠ 0.
You can also Factorise these equations.
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’
 x2 – y2 = (x + y)(x – y) 1E

22. Algebra and Functions 1E Factorising Quadratics
A Quadratic Equation has the form;
ax2 + bx + c
Where a, b and c are constants and a ≠ 0.
You can also Factorise these equations.
Examples d)
The 2 numbers in brackets must:
Multiply to give ‘c’
Add to give ‘b’ 1E

23. Algebra and Functions 1E Factorising Quadratics
A Quadratic Equation has the form;
ax2 + bx + c
Where a, b and c are constants and a ≠ 0.
You can also Factorise these equations.
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… 1E

24. Algebra and Functions Expand the following pairs of brackets
(x + 3)(x + 4)
 x2 + 3x + 4x + 12
 x2 + 7x + 12
(2x + 3)(x + 4)
 2x2 + 3x + 8x + 12
 2x2 + 11x + 12 x
+ 3 x x2
+ 3x
When an x term has a ‘2’ coefficient, the rules are different…
+ 4
+ 4x
+ 12
2 of the terms are doubled
 So, the numbers in the brackets add to give the x term, WHEN ONE HAS BEEN DOUBLED FIRST 2x
+ 3 x
2x2
+ 3x
+ 4
+ 8x
+ 12

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

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

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

28. Teachings for Exercise 1F

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

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

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

32. Teachings for Exercise 1G

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

34. Teachings for Exercise 1H

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

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

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

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!