# Cubic curves II: cubic functions and their graphs

## General cubic functions

The general cubic function has the form \(\normalsize{y=ax^3+bx^2+cx+d}\) and has a somewhat different shape to the standard cubic \(\normalsize{y=ax^3}\). We discuss the general form of such functions, and the relation with any zeroes it might have: there are at most three zeroes, but a general cubic need not have all three zeroes, even approximately.## More general cubic curves

Isaac Newton investigated more general cubic curves, given by arbitrary degree-three polynomials in \(\normalsize{x}\) and \(\normalsize{y}\). Earlier, both Fermat and Descartes had investigated special cubics. The theory of these kinds of curves is rich and full of surprises.## Zeroes and coefficients

The zeroes of a polynomial, if they are known, and the coefficients of that polynomial are two different sets of numbers that have interesting relations. If we know the zeroes, then we can write down algebraic expressions for the coefficients. Going the other way is much harder and cannot be done in general. But in special cases, it can be done.Q1(M): What is the factorisation of \(\normalsize{f(x)=x^3+2x^2-5x-6}\)?

## Answers

\[\Large{f(x)=x^3+2x^2-5x-6=(x+1)(x-2)(x+3)}.\]A1.The factorisation is

#### Maths for Humans: Inverse Relations and Power Laws

## Our purpose is to transform access to education.

We offer a diverse selection of courses from leading universities and cultural institutions from around the world. These are delivered one step at a time, and are accessible on mobile, tablet and desktop, so you can fit learning around your life.

We believe learning should be an enjoyable, social experience, so our courses offer the opportunity to discuss what you’re learning with others as you go, helping you make fresh discoveries and form new ideas.

You can unlock new opportunities with unlimited access to hundreds of online short courses for a year by subscribing to our Unlimited package. Build your knowledge with top universities and organisations.

Learn more about how FutureLearn is transforming access to education