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".

A basic introduction to differentiation

Using the chain rule with polynomials

More work on differentiation with fractional coefficients

More work on differentiation with fractional coefficients

Quotient and remainder, polynomial division.

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

A basic introduction to differentiation

Using the chain rule with polynomials

Quotient and remainder, polynomial division.

Quotient and remainder, polynomial division.

More work on differentiation with fractional coefficients

Using the chain rule with polynomials and negative powers.

More work on differentiation with fractional coefficients and powers

More work on differentiation with fractional coefficients.

Using the chain rule with polynomials and negative powers

A basic introduction to differentiation

A basic introduction to differentiation

Using the chain rule with polynomials

More work on differentiation with fractional coefficients

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

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. 

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

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

Identify whether or not an expression is a polynomial. From HELM Book 2.7.1.