## Algebraic Manipulation 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 Expanding Brackets Topic 10 Polynomials Topic 2 Factorising Expressions Topic 11 Simultaneous Equations Topic 3 Exponents Topic 12 Inequalities Topic 4 Algebraic Fractions Topic 13 Sequences Topic 5 Logarithms Topic 14 Arithmetic Sequences Topic 6 Linear Equations Topic 15 Geometric Sequences Topic 7 Formulae Topic 16 Sigma Notation Topic 8 Quadratic Equations Topic 17 Arithmetic Series Topic 9 Nature of Roots 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   Example 2 Simplify   Example 3 Simplify   ### 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)