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Chapter 4. Polynomials
70
conjugate of conjugate
¯ z = z for all z ∈ C;
multiplicativity of absolute value
|wz|=|w||z| for all w, z ∈ C.
In the next result, we need to think of a polynomial with real coef-
ficients as an element of P(C). This makes sense because every real
number is also a complex number.

A polynomial with real 4.10 Proposition: Suppose p is a polynomial with real coefficients.
¯
coefficients may have If λ ∈ C is a root of p, then so is λ.
no real roots. For
example, the Proof: Let
m
2
polynomial 1 + x has p(z) = a 0 + a 1 z +· · ·+ a m z ,
no real roots. The
where a 0 ,...,a m are real numbers. Suppose λ ∈ C is a root of p. Then
failure of the
fundamental theorem a 0 + a 1 λ +· · ·+ a m λ m = 0.
of algebra for R
accounts for the Take the complex conjugate of both sides of this equation, obtaining
differences between
¯
¯ m
operators on real and a 0 + a 1 λ +· · ·+ a m λ = 0,
complex vector spaces,
where we have used some of the basic properties of complex conjuga-
as we will see in later
¯
tion listed earlier. The equation above shows that λ is a root of p.
chapters.
We want to prove a factorization theorem for polynomials with real
coefficients. To do this, we begin by characterizing the polynomials
with real coefficients and degree 2 that can be written as the product
of two polynomials with real coefficients and degree 1.

Think about the 4.11 Proposition: Let α, β ∈ R. Then there is a polynomial factor-
connection between the ization of the form
quadratic formula and
2
this proposition. 4.12 x + αx + β = (x − λ 1 )(x − λ 2 ),
2
with λ 1 ,λ 2 ∈ R, if and only if α ≥ 4β.

Proof: Notice that
α α 2
2
2
4.13 x + αx + β = (x + ) + (β − ).
2 4