2.2 Polynomials 55ImP′(w)
P(w)=
∑nk= 1Im1
w−zk> 0.
This shows thatwis not a zero ofP′(z)and so all zeros ofP′(z)lie in the upper half-plane.
The theorem is proved.
173.Leta 1 ,a 2 ,...,anbe positive real numbers. Prove that the polynomialP(x)=
xn−a 1 xn−^1 −a 2 xn−^2 −···−anhas a unique positive zero.
174.Prove that the zeros of the polynomial
P(z)=z^7 + 7 z^4 + 4 z+ 1
lie inside the disk of radius 2 centered at the origin.175.Fora =0 a real number andn>2 an integer, prove that every nonreal rootzof
the polynomial equationxn+ax+ 1 =0 satisfies the inequality|z|≥n√
1
n− 1.176.Leta∈Candn≥2. Prove that the polynomial equationaxn+x+ 1 =0 has a
root of absolute value less than or equal to 2.
177.LetP(z)be a polynomial of degreen, all of whose zeros have absolute value 1 in
the complex plane. Setg(z)=P(z)zn/ 2. Show that all roots of the equationg′(z)= 0
have absolute value 1.
178.The polynomialx^4 − 2 x^2 +ax+bhas four distinct real zeros. Show that the
absolute value of each zero is smaller than
√
3.
179.LetPn(z),n≥1, be a sequence of monickth-degree polynomials whose coefficients
converge to the coefficients of a monickth-degree polynomialP(z). Prove that for
any>0 there isn 0 such that ifn≥n 0 then|zi(n)−zi|<,i= 1 , 2 ,...,k, where
zi(n)are the zeros ofPn(z)andziare the zeros ofP(z), taken in the appropriate
order.
180.LetP(x)=anxn+an− 1 xn−^1 +···+a 0 be a polynomial with complex coefficients,
witha 0 =0, and with the property that there exists anmsuch that
∣
∣∣
∣
am
a 0∣
∣∣
∣≥
(
n
m)
.
Prove thatP(x)has a zero of absolute value less than 1.181.For a polynomialP(x)=(x−x 1 )(x−x 2 )···(x−xn), with distinct real zeros
x 1 <x 2 <···<xn, we setδ(P(x))=mini(xi+ 1 −xi). Prove that for any real
numberk,
δ(P′(x)−kP (x)) > δ(P (x)),
whereP′(x)is the derivative ofP(x). In particular,δ(P′(x)) > δ(P (x)).