Number & Algebra

Binomial expansion

Manipulation of surds

Rationalise denominators

Quadratic formula

Complete the square

Factor theorem

Factor theorem solves cubic equation (ax3+bx2+cx+d=0)

Given that (x – 5) is a factor of x3 – 8x2 + 17x – 10, work out the quadratic factor.

Domain & range of a function

Composite functions

Express (5x + 6)4 in the form fg(x), stating the expressions corresponding to f(x) and g(x).

Starting with x gives the flow chart x → 5x + 6 → (5x + 6)4

This is the same as x → g(x) → fg(x) giving g(x) = 5x + 6, f(x) = x4 so fg(x) = (5x + 6)4

Given that f(x) = x + 1 and g(x) = x2 – 1. Find the composite function of fg(x) and gf(x).

Inverse functions

Graphs of exponential functions

Graphs of quadratic functions

General formula of quadratic function

Linear inequalities

When solving a linear inequality, if you multiply or divide by a negative, you reverse the sign.

Quadratic inequalities

To solve a quadratic inequality, find the critical values, then use a sketch to interpret these values as solution regions.

Index laws

Disguised quadratic equations

Algebraic proof

Sequences
Number sequence
A number sequence is a set of numbers listed in a specific order, where the numbers can be found by a specific rule.

 

Arithmetic progression (AP)

An arithmetic progression is a linear sequence in which the next term is obtained by adding a fixed common difference d, to the previous term

  • un = the nth term
  • a = the 1st term
  • n = the number of terms
  • d = common difference

 

Sum of arithmetic progression (AP)

The sum of the first n terms of an AP with first term a and common difference d is given by

  • Sn = the sum of the first n terms
  • a = the 1st term
  • n = the number of terms
  • d = common difference

 

The difference method

Use of a difference method to find the formula for a linear sequence, a quadratic sequence or a cubic sequence is required.

For the linear sequence, the first differences are constant.

For the quadratic sequence, the second differences are constant.

For the cubic sequence, , the third differences are constant.

Limiting value

Find the limiting value of