Page 78 - Linear Algebra Done Right

Chapter 4. Polynomials
64
Degree
Recall that a function p: F → F is called a polynomial with coeffi-
cients in F if there exist a 0 ,...,a m ∈ F such that
2
p(z) = a 0 + a 1 z + a 2 z +· · ·+ a m z m
for all z ∈ F.If p can be written in the form above with a m = 0, then we
say that p has degree m. If all the coefficients a 0 ,...,a m equal 0, then
When necessary, use we say that p has degree −. For all we know at this stage, a polynomial
the obvious arithmetic may have more than one degree because we have not yet proved that
with −. For example, the coefficients in the equation above are uniquely determined by the
− <m and function p.
− + m =−∞ for Recall that P(F) denotes the vector space of all polynomials with
every integer m. The 0 coefficients in F and that P m (F) is the subspace of P(F) consisting of
polynomial is declared the polynomials with coefficients in F and degree at most m. A number
to have degree − so λ ∈ F is called a root of a polynomial p ∈P(F) if
that exceptions are not
needed for various p(λ) = 0.
reasonable results. For
Roots play a crucial role in the study of polynomials. We begin by
example, the degree of
showing that λ is a root of p if and only if p is a polynomial multiple
pq equals the degree of
of z − λ.
p plus the degree of q
even if p = 0.
4.1 Proposition: Suppose p ∈P(F) is a polynomial with degree
m ≥ 1. Let λ ∈ F. Then λ is a root of p if and only if there is a
polynomial q ∈P(F) with degree m − 1 such that

4.2 p(z) = (z − λ)q(z)

for all z ∈ F.

Proof: One direction is obvious. Namely, suppose there is a poly-
nomial q ∈P(F) such that 4.2 holds. Then

p(λ) = (λ − λ)q(λ) = 0,

and hence λ is a root of p, as desired.
To prove the other direction, suppose that λ ∈ F is a root of p. Let
a 0 ,...,a m ∈ F be such that a m = 0 and

2
p(z) = a 0 + a 1 z + a 2 z +· · ·+ a m z m