Given two complex numbers, find by inspection the one that is a root of a given quartic real polynomial and hence find the other roots. 

Factorising polynomials using the highest common factor

Use Matlab (or Python) to fit a cubic polynomial to data, and determine where the x-axis is crossed.

Finding the zero of a non-polynomial function

Differentiation of  polynomials, cos, sin, exp, log functions. Product, quotient and chain rules.

rebelmaths

Factorising polynomials using the highest common factor.

Adapted from 'Factorisation' by Steve Kilgallon.

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Find the remainder when dividing two polynomials, by algebraic long division.

Explore mode question guiding you through the steps of the "I can name your polynomial" trick.

Calculating the integral of a function of the form $a_1x^{b_1}+a_2x^{b_2}+a_3x^{b_3}$ using a table of integrals. 

Differentiate a polynomial expression involving coefficients and, negative and fractional indices. 

Find the derivative of a function of the form $y=ax^b$ using a table of derivatives.

Multiple choice question. Given a randomised polynomial select the possibe ways of writing the domain of the function.

Understanding the general facts about polynomials of degree n.

Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$.

Example of an explore mode question. Student is given a 3x3 matrix and is asked to find the characteristic polynomial and eigenvalues, and then eigenvectors for each eigenvalue. The part asking for eigenvectors can be repeated as often as the student wants, to be used for different eigenvalues.

Assessed: calculating characteristic polynomial and eigenvectors.

Feature: any correct eigenvalue will be recognised by the marking algorithm, even multiples of the obvious one(s) (which can be read off from the reduced row echelon form)

Randomisation: Not randomised, just using particular matrices. I am still working on how to randomise this for 3x3; a randomised 2x2 version exists. I have several different versions for 3x3 (not all published yet), so I could make a random choice between these in a test.

The implementation uses linear algebra functions such as "find reduced echelon form" or "find kernel of a reduced echelon form", from the extension "linalg2".

Using the chain rule with polynomials.

Quotient and remainder, polynomial division.

A basic introduction to differentiation

Quotient and remainder, polynomial division.

Using the chain rule with polynomials

Quotient and remainder, polynomial division.

Factorising polynomials using the highest common factor

A basic introduction to differentiation

More work on differentiation with fractional coefficients

A basic introduction to differentiation

More work on differentiation with fractional coefficients

Using the chain rule with polynomials

Quotient and remainder, polynomial division.