1. Introduction: One Step Up From the Parabola
Let me start with something you already know.
A quadratic polynomial is one whose highest power of x is 2 — like x² − 5x + 6. You have seen its graph: a U-shaped curve called a parabola, and it can meet the x-axis at most twice. So a quadratic has at most two zeroes. A zero of a polynomial is a value of x that makes the polynomial equal to 0.
Now I am going to push the power up by one. What happens when the highest power is 3? Something like x³ − 6x² + 11x − 6.
Stop scrolling. Try it in your head before reading on. A quadratic gave at most two zeroes. If the highest power is now 3, what is your best guess for the most zeroes this can have?
If you guessed three, you are thinking exactly right. A polynomial whose highest power is 3 is called a cubic polynomial, and the big idea of this lesson is: it has at most three zeroes, and those three zeroes are tied to the polynomial's numbers — its coefficients — by three neat rules.
By the end you will be able to recognise a cubic on sight, and given its three zeroes, predict its coefficients without expanding anything.