algebra.jpgIn 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
Topic 10
Topic 2
Topic 11
Topic 3
Topic 12
Topic 4
Topic 13
Topic 5
Topic 14
Topic 6
Topic 15
Topic 7
Formulae
Topic 16
Topic 8
Topic 17
Arithmetic Series
Topic 9
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)

= (x2 − 2x − 3)(x − 3)

= x3 − 2x2 − 3x − 3x2 + 6x + 9

= x3 − 5x2 + 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 Y12_Algebraic_Manipulation_01.gif
Y12_Algebraic_Manipulation_02.gif

Y12_Algebraic_Manipulation_03.gif

Example 2 Simplify Y12_Algebraic_Manipulation_04.gif
Y12_Algebraic_Manipulation_05.gif

Y12_Algebraic_Manipulation_06.gif

Example 3 Simplify Y12_Algebraic_Manipulation_07.gif
Y12_Algebraic_Manipulation_08.gif

Y12_Algebraic_Manipulation_09.gif

 

Sum and Difference of Two Cubes

The following two factorisations are sometimes useful:

The sum of two cubes:

x3 + y3 = (x + y)(x2 − xy + y2)

The difference of two cubes:

x3 – y3 = (x –y)(x2 + xy + y2)

Example

64a3 − 27b3 = (4a)3 − (3b)3

                    = (4a − 3b)(16a2 + 12ab + 9b2)