UCL Online STEP and AEA Preparation Sessions
Session 5: Quadratics, cubic and other polynomials
Factorising and expanding polynomials; sketching graphs of
polynomial functions; using techniques such as ‘completing the
square’; using the factor and remainder theorem and employing
substitutions to allow the use of polynomial techniques will all be
considered in this session.
It’s also useful to have knowledge of the relationship between the
roots and coefficients of polynomial equations.
For example:
If  ,  ,  are roots of the polynomial x3  ax2  bx  c  0 then
a  (     ), b       , c  
The fundamental theorem of algebra will be discussed as well as
ways to identify the number of real roots a polynomial equation
has.
There will be a discussion of using the first derivative to determine
the nature of stationary points of polynomial functions.
Problems involving expanding algebraic expressions involving
polynomials, as in the example below, will be considered. This
involves revisiting ideas in the last chapter to do with counting and
placement:
Example problem
Calculate the coefficient of x40 in the expansion of
x
2
 1
14
x
4
 x 2  1
4