Concept preview

What you will be able to do

1. 01Predict end behaviour from degree and leading coefficient
2. 02Use multiplicity to decide whether a graph crosses or touches at a root
3. 03Connect factored form, roots, turning points, and the shape of the graph

A polynomial function is a sum of terms $a_n x^{n} + a_{n-1}x^{n-1} + \dots + a_1 x + a_0$ where the exponents are whole numbers. The degree $n$ is the highest exponent, and $a_n$ is the leading coefficient. Everything about the graph's large-scale shape — how many times it can turn, how many roots it can have, and what its arms do far out — is controlled by just these two numbers, so we start there.