Advanced book on Mathematics Olympiad

52 2 Algebra

2.2.3 The Derivative of a Polynomial.............................

This section adds some elements of real analysis. We remind the reader that the derivative
of a polynomial

P(x)=anxn+an− 1 xn−^1 +···+a 1 x+a 0

is the polynomial

P′(x)=nanxn−^1 +(n− 1 )an− 1 xn−^2 +···+a 1.

Ifx 1 ,x 2 ,...,xnare the zeros ofP(x), then by the product rule,

P′(x)

P(x)

=

1

x−x 1

+

1

x−x 2

+···+

1

x−xn

.

If a zero ofP(x)has multiplicity greater than 1, then it is also a zero ofP′(x), and the
converse is also true. By Rolle’s theorem, if all zeros ofP(x)are real, then so are those
ofP′(x). Let us discuss in detail two problems, the second of which is authored by
R. Gologan.

Example.Find all polynomialsP(x)with the property thatP(x)is a multiple ofP′′(x).

Solution.LetP(x)=Q(x)P′′(x), withQ(x)a polynomial that is necessarily quadratic.
Since the multiplicity of a zero ofP(x)is strictly greater than the multiplicity of the same
zero inP′′(x), it follows thatP(x)has at most two distinct zeros, and these must be zeros
ofQ(x). So letP(x)=α(x−a)k(x−b)n−k.
Ifa =b, thenaandbare both zeros ofQ(x),soP′′(x)=n(n− 1 )α(x−a)k−^1 (x−
b)n−k−^1 andQ(x)= n(n^1 − 1 )(x−a)(x−b). But this cannot happen unlessk− 1 =
n−k− 1 =0, for if a number is a zero of bothP(x)andP′′(x), then the difference
between the multiplicities of this zero in the two polynomials is 2.
Ifa =b, thenP(x)=α(x−a)n,n≥2,isa multiple ofP′′(x). The answer
to the problem consists of all quadratic polynomials and all polynomials of the form
P(x)=α(x−a)n,n≥2. 

Example.LetP(x)∈Z[x]be a polynomial withndistinct integer zeros. Prove that the
polynomial(P (x))^2 +1 has a factor of degree at least 2n+ 21 that is irreducible overZ[x].

Solution.The statement apparently offers no clue about derivatives. The standard ap-
proach is to assume that

(P (x))^2 + 1 =P 1 (x)P 2 (x)···Pk(x)

is a decomposition into factors that are irreducible overZ[x]. Lettingx 1 ,x 2 ,...,xnbe
the integer zeros ofP(x), we find that