1. Introduction: The Graph Already Knows the Answer
You have solved equations like x² − 5x + 6 = 0 before. You factorised, you got x = 2 and x = 3, and you moved on. Those two numbers have a name — they are the zeroes of the polynomial x² − 5x + 6.
Here is a question you may never have asked. What do those zeroes look like? If you drew the graph of y = x² − 5x + 6 on a sheet of graph paper, where would the answers 2 and 3 actually sit?
Stop scrolling. Try it in your head before reading on. If a number is a zero of the polynomial, what must the graph be doing at that point?
It turns out you do not always need to factorise at all. The graph hands you the zeroes directly — you just have to know where to look. In this section you will learn to read zeroes off a curve, and to count how many a polynomial can have just from the shape of its graph.
By the end you will be able to look at any graph and point straight to the zeroes, and you will be able to say in advance how many zeroes a line, a parabola, or a cubic can possibly have.