# Chapter one

Mathematical skills

## Section four: Roots of polynomials

This chapter is on the theme of supporting mathematics that are needed for engineering problem solving, analysis, design and evaluation.

It is implicit that for many engineering topics, students have core competence in some mathematical topics such as polynomials, roots, complex numbers, exponentials, logarithms and Laplace.

This section focuses on roots of polynomials. Students need to be skilled at moving between polynomial roots and the coefficients, and recognising patterns which facilitate estimation.

### 1. Roots from a plot

Lectures aimed at understanding what is a root, links to polynomial factors and easy methods for factorising by hand. This video is an introduction which focuses on understanding what we mean by a root.

A talk through video is on YouTube.

### 2. What is a factor?

This video is an introduction which focuses on understanding what we mean by a factor.

A talk through video is on YouTube.

### 3. First order

This video is an introduction which focuses on first order polynomials.

A talk through video is on YouTube.

### 4. Second order

This video is an introduction which focuses on second order polynomials.

A talk through video is on YouTube.

### 5. Higher order

This video is an introduction which focuses on high order polynomials.

A talk through video is on YouTube.

### 6. Factorising high order polynomials

This video focuses on numerical examples of how to factorise a polynomial.

A talk through video is on YouTube.

### 7. Poles, zeros and transfer functions

An introduction to concepts of poles and zeros for transfer functions and the meaning of left half plane (LHP) and right half plane (RHP).

A talk through video is on YouTube.

### 8. Worked examples and inequalities

This video contains a number of worked examples.

A talk through video is on YouTube.

### 9. Assessing stability and the Routh array

A key reason for computing poles is due to the strong link with system behaviours and in particular stability. This video looks at simple, back of the envelope calculations that can be used to determine whether the roots are all in the LHP, or not.

Talk through video is on YouTube.

### 10. Tutorial sheet and use of MATLAB

This video gives a number of tutorial examples and the worked solutions, including factorisation and the RHP. It also illustrates how MATLAB can be used both to validate your own working and to determine roots when the computations are difficult by hand.

Talk through video is on YouTube.