Mathematics: Analysis and Approaches · Topic 2: Functions

AHL2.12 — Polynomial functions, roots and factors

Mathematics: Analysis and Approaches · SL / HL · syllabus-mapped notes

1. Polynomials, zeros, roots and factors

   Polynomial functions $P(x)=a_nx^n+\dots+a_0$, their graphs, and the link between zeros, roots and factors.

2. The factor and remainder theorems

   $(x-a)$ is a factor iff $P(a)=0$; the remainder on dividing by $(x-a)$ is $P(a)$.

3. Sum and product of the roots

   For $\sum a_rx^r=0$: sum of roots $=-\dfrac{a_{n-1}}{a_n}$, product $=\dfrac{(-1)^na_0}{a_n}$.