# polynomials

## Introduction

The first section of this course is all about solving polynomial equations. You're familiar with solving quadratic equations, both by factorising (where appropriate) and by completing the square, and these are two techniques which we will make extensive use of. Your first task is to review these techniques. You will also be familiar with the quadratic formula for solving these equations, and the motivation for the rest of this section is to try to extend the idea of a general method to cubic and quartic equations. Each mini-section below should take between 30 and 50 minutes to work through, though some are shorter, and you may find that some of the techniques are tricky to grasp and you may need to re-watch the videos. Don't worry overly if you do find this material difficult! It's intended to be challenging, and one of the key purposes is to help develop and improve your algebraic skills.

Work through this sheet, and after you've completed it, check your work with these solutions.

## 2. Hidden quadratics - part one

A very useful technique for solving some equations is to recognise that they're really quadratics in disguise. Watch the video below and then work through questions 1-10 on the first page of the document here. After you've attempted them all, check your work with the solutions.

The notes I made with the video are here.

## 3. Hidden quadratics - part two

The video below shows a nice trick for solving some types of quartic equation by making them into a hidden quadratic. Watch the video and read through the accompanying notes; then work through the questions on the second page of the questions document.

Solutions

## 4. Cubics - part one

A short section here - mainly a bit of algebra practice. Read through the notes and have a go at the first couple of questions. Watch the linked video if you need to, and then complete the other questions.

Notes

Solutions

## 5. Cubics - part two

This is a lovely method for solving certain cubic (third power) equations - ones which are known as depressed cubics. The method was derived in the mid-16th century by the Italian mathematician Tartaglia, who seemed to spend a good deal of his time setting and answering mathematical questions with his correspondents, not least his rival Cardano. The method builds on what we've done so far, and involves solving the cubics by making a neat substitution which leads to a hidden quadratic. An interesting side-product of the method, which has proved spectacularly important in mathematics, is the idea of complex numbers.

Watch the video below and then complete the questions.

Questions

## 6. Cubics - part three

Another short section, with an adaptation of Tartaglia's method that allows us to find a solution to any cubic equation. This part of the method was Cardano's contribution to the subject. Watch the video below, then have a go at the questions below.

Questions

Solutions

## 7. Quartics - optional section

A method that brings everything together, and allows us to solve any quartic equation. It uses the method for solving cubics, which in turn relies on being able to solve quadratics. Be warned: there's a lot of algebra involved!

Work through this document. Watch the video below where indicated.

An interesting fact is that it is not possible - in general terms - to solve any higher degree polynomial than a quartic. It's possible to find a 'cubic formula' and a 'quartic formula' but not a 'quintic formula', a fact which was first proved by the Norwegian mathematician Niels Abel, and later - and more elegantly - by the French  mathematician Évariste Galois. Both these mathematicians had tragically short lives: Abel died aged 26 and Galois died (after a duel) aged only 20. Find out more about the maths behind the proof here.