1. ALGEBRAIC EXPRESSIONS
Real numbers
Surds lie between integers
Rounding-off Integers
Products
Factorization: Common Factor
Factorization: Difference of Squares
Factorization: Trinomials
Factorization: Grouping in Pairs
Factorization: Cubes
Algebraic Fractions
Algebraic Language

2. REAL NUMBERS Rational Numbers
A rational number is a number that can be expressed in the form where
and where a and b are integers.

3. Examples
a) Integers
e.g. 5 can be written as where 5 and 1 are integers.
b) Mixed fractions
e.g.
c) Terminating decimals
e.g. 0,25 =
d) Recurring decimals have an infinite pattern & can be expressed as a fraction
e.g. 0, 3 = 0, ; 0, 12 = 0,

4. Converting Recurring Decimals to Fractions
E.g. a) Show that 0,3 is rational.
Let x = 0,
l0x = 3, (multiply both sides by 10)
l0x - x = 3, – 0, (subtract equations)
9x = 3,
9x = 3
x = 3 … a rational number!
E.g. b) Show that 0, 12 is rational.
Let x = 0,
100x = 12, (multiply both sides by 100)
99x = 12, (subtract equations)
x = … a rational number!

5. EXERCISE 1. Are these numbers rational and why? (a) (b) (c) 6 (d) - 3
(f) 1, 4142

6. 2. Show that the following recurring decimals
are rational:
(a) 0, 4
(b) 0, 21
(c) 0, 14
(d) 19, 45
(e) 0, 124
(f) 0, 124

7. Irrational Numbers in Circles & Squares
Numbers that cannot be written in the form
where
Therefore recurring numbers that neither terminate nor recur with a pattern
E.g. a) 5,739129…
b) -4, … c)
Irrational Numbers in Circles & Squares

8. The Number
is the ratio of the circumference of a circle to its diameter
is = 3, ….
However, can be approximated as an
improper fraction
Rounding-off π
π as a Rational Number

9. EXERCISE 1
State whether the following numbers are rational or irrational:
(a) 8 (b) (c) 7 (d)
(e) (f) (g) (h)
(i) - 0, (j) 0, 42 (k)
(l) 0, …

10. EXERCISE 2 Classify numbers by placing ticks in the appropriate columns:
Real
Rational
Integer
Whole
Natural
Irrational
- 3
0, 3
8, 23647

11. SURDS LIE BETWEEN INTEGERS
E.g. Determine without the use of a calculator,
between which 2 integers lies.
Find an integer smaller and bigger than 11 that can be square rooted … 9 and 16
Now create an inequality … 9 < 11 < 16
Square root all integers …
Solve …
Check using a calculator …

12. EXERCISE
Without using a calculator, determine between which two integers the following irrational numbers lie:
(a)
(b)
(c)
(d)
(e)
(f)

13. ROUNDING-OFF INTEGERS
If it is > 5 or = 5 … round up
If it is < … round down
Remember! If you are rounding-off to 2 decimal places, the third decimal place determines whether you round up or down etc.
E.g. (a) 2, (2 d.p.) … Answer: 2, 31
(b) 0, (3 d.p.) … Answer: 0, 778
(c) 245, (4 d.p.) … Answer: 245,1359
Rounding-off Numbers

14. EXERCISE 1
Round off the following numbers to the number of decimal places indicated:
(a) 9, (3 decimal places)
(b) 67, (2 decimal places)
(c) 4, (4 decimal places)
(d) 17, (5 decimal places)
(e) 79, (3 decimal places)
(f) 34, (4 decimal places)
(g) 5, (5 decimal places)

15. EXERCISE 2
Simplify and round-off to the number of decimal places indicated:
(a) (3 decimal places)
(b) (4 decimal places)
(c) (2 decimal places)
(d) (5 decimal places)
(e) (2 decimal places)

16. PRODUCTS E.g. x (y + z) = xy + xz
- Multiply each term inside the bracket by the number outside the bracket
E.g. (a + b)(c + d) = ac + ad + bc + bd
- This is done by using the FOIL method
The Product Game

17. Squaring a Binomial Example
Examples
Expand and simplify the following:
(a) (x + 2) (x + 3)
(b) (x + 2) (x² + x - 1)
Squaring a Binomial Example

18. (c) x(x²-2xy+3y²) - 2y(x² -2xy+3y²)
(d) (a – 3b) (a – 3b)²

19. Exercise 1: Simplify: (a) (x + 3)(x - 3) (b) (x - 6)(x + 6)
(c) (2x - l)(2x + l)
(d) (4x + 9)(4x - 9)
(e) (3x - 2y)(3x + 2y)
(f) (4a³ b + 3)(4a³ b - 3)
(g) (2x – 3 + y)(2x – 3 – y)
(h) (1 – a )(1 – a )(1 + a)

20. Exercise 2 Simplify: (a) 2x(3x - 4y)² - (7x - 2xy)
(b) (5y + 1)² - (3y + 4)( y)
(c) (2x + y) - (3x - 2y) + (x - 4y)(x + 4y)
(d) (8m - 3n)(4m + n) - (n - 3m)(n + 3m)
(e) (3a + b)(3a - b)(2a + 5b)

21. Exercise 3 Simplify: (a) (x + 1)(x² + 2x + 3) (b) (x - 1)(x² - 2x + 3)
(c) (2x + 4)(x² - 3x + 1)
(d) (2x - 4)(x² - 3x + 1)
(e) (3x-y)(2x² + 4xy – y² )
(f) (3x - 2y)(9 x² + 6xy + 4 y² )
(g) (3x + 2y)(9x² - 6xy + 4y² )
(h) (2a + 3b)²
(i) (2a² - 3b)²

22. FACTORIZATION: Common Factor
The Factor Game
The golden rule of factorization is to always look for the highest common factor first:
Basic examples
Common Factor with Variables
Complex example
e.g. a(x-y) – 2(x-y)² = (x-y)[a-2(x-y)]
= (x-y)(a-2x+2y)

23. Common Brackets Exercise:
(a) (b)
(c) (d)
(e) (f)
(g) (h)
(i) (j)
(k) (l)
(m) (n)
(o)

24. FACTORIZATION: Difference of Squares
There must be 2 terms that you can take the square-root of and a minus sign.
Basic examples a) b)
Difference of Squares Example

25. Complex examples a) b) c)

26. Complex Exercise
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(n)

27. FACTORIZATION: Trinomials
Make sure you know your times-tables and factors!
E.g. a)
Factors of 8: 1 x 8 or 4 x 2
The middle term (6a) is obtained by adding the factors of 8 … = 6
Therefore:
E.g. b)
First take out common factor!
The middle term (7x) is obtained by adding the factors of 8 … = -7
Trinomial with Common Factor

28. Visualizing Factorization
Note:
If the sign of the last term of a trinomial is positive, the signs in the brackets are the same
i.e. (… - …)(… - …) or (… + …)(… + …)
If the sign of the last term of a trinomial is negative, the signs in the brackets are different, i.e. both positive or both negative
i.e. (… + …)(… - …) or (… - …)(… + …)
Visualizing Factorization

29. Basic Exercise Factorize fully : (a) (b) (c) (d) (e) (f) (g) (h)
(i) (j)
(k) (l)
(m) (o)

30. Complex Exercise
Factorize fully:
(a)
(b)
(c)
(d)
(e)

31. More advanced trinomials
E.g. a)
Step 1: Check for the HCF … none
Step 2: Write down the brackets and the factors of the first term and the factors of the last term …
(7p 1)(3p )
Step 3: Now multiply the innermost and the outermost terms …
Step 4: To find the middle term p + 28p = + 25p
Step 5: Complete the factors … (7p – 1)(3p + 4)
Note! This method involves trial and error and you need to keep t
trying different options until you get ones that will work.

32. E.g. b) Step 1: Check for the HCF … none
Step 2: Write down the brackets and the factors of the first term and the factors of the last term …
(12a b)(2a b)
Step 3: Now multiply the innermost and the outermost terms … 2ab x 12ab
Step 4: To find the middle term ab + 2ab = - 10ab
Step 5: Complete the factors … (12a + b)(2a – b)

33. Advanced Exercise 1. Factorize fully: (a) (b) (c) (d) (e) (f) (g) (h)
(i) (j)
(k) (l)

34. FACTORIZATION: Grouping in pairs
Group terms with common factors or similar brackets!
E.g.
Group the terms that look similar
(i.e. those that could potentially have
common factors)
Factorize each pair separately and then take out the common bracket:

35. Switch-arounds “taking out a negative”
- x + y = - (x - y) and – x – y = - (x + y)
E.g. a) Factorize

36. b) Factorize:
(c) Factorize:

37. Exercise
Factorize:
(a)
(b)
(c)
(d)
(e)

38. x³ + y³ = (x+y)(x² - xy + y²)
FACTORIZATION: Cubes
Sum of Cubes
x.x²=x³ y.y²=y³
x³ + y³ = (x+y)(x² - xy + y²)
Take the cube root
of each term
Times factors of first bracket to get middle term
Sum of Cubes Example

39. x³ - y³ = (x-y)(x² + xy + y²)
Difference of Cubes
x.x²=x³ y.y²=y³
x³ - y³ = (x-y)(x² + xy + y²)
Take the cube root
of each term
Times factors of first bracket to get middle term
Difference of Cubes Example
Visualization of Factorizing a Cubic Expression

40. ALGEBRAIC FRACTIONS
Simplify the following expressions:
(a)
(b)

41. Simplifying Basic Algebraic Expressions
Whenever the numerator contains two or more terms, factorize the expression in the numerator and simplify
(c)
(d)
Simplifying Basic Algebraic Expressions

42. EXERCISE 1 Simplify the following: (a) (b) (c) (d) (e) (f) (g) (h)
(i) (j)

43. EXERCISE 2
Simplify
(a) (b)
(c) (d)
(e) (f)

44. More Advanced Algebraic Fractions
Examples a) b)
Simplifying Complex Algebraic Expressions

45. Exercise
Simplify: a) b) c) d)