In Year 11 (NZ Year 12), the following topics were extensively covered and it is assumed that students will have a thorough understanding of the content covered below.
Topic 1


Formulae  
Topic 17

Arithmetic Series  
Topic 18

Geometric Series 
The next few topics in the Mathematics with Calculus course will concentrate on more advanced examples from the above topics.
Expanding Brackets
 If there are three or more brackets , the best method is to multipy two together first and then the third etc.
 If surds are involved then remember √a x √b = √ab
Example 1

Expand (x + 1)(x − 3)^{2} 
(x + 1)(x − 3)^{2}

= (x + 1)(x − 3)(x − 3) = (x^{2} − 2x − 3)(x − 3) = x^{3} − 2x^{2} − 3x − 3x^{2} + 6x + 9 = x^{3} − 5x^{2} + 3x + 9 
Example 2

Expand (√2 + 3)(√2 − 3) 
(√2 + 3)(√2 − 3)

= 2 + 3√2 − 3√2 − 9 = 2 − 7 = 5 
Simplifying Expressions
 The laws of indices apply
 Look for common factors
 Like factors from the numerator and denominator can be cancelled.
Example 1  Simplify  
Example 2  Simplify  
Example 3  Simplify 
Sum and Difference of Two Cubes
The following two factorisations are sometimes useful:
The sum of two cubes: x^{3} + y^{3} = (x + y)(x^{2} − xy + y^{2}) 
The difference of two cubes: x^{3} – y^{3} = (x –y)(x^{2} + xy + y^{2}) 
Example
64a^{3} − 27b^{3} = (4a)^{3} − (3b)^{3}
= (4a − 3b)(16a^{2} + 12ab + 9b^{2})