
Chapter 3
Factorisation of Polynomials
Not only is there an algebra of polynomials: there is an arithmetic. That is, there
are notions analogous to the integer-based concepts of divisibility, primes, prime
factorisation, and highest common factors. These notions are essential for any serious
understanding of polynomial equations, and we develop them in this chapter.
Mathematicians noticed early on that if f is a product gh of polynomials of
smaller degree, then the solutions of f (t) = 0 are precisely those of g(t) = 0 together
with those of h(t) = 0. For example, to solve the equation
t
3
−6t
2
+ 11t −6 = 0
we can spot the factorisation (t −1)(t −2)(t