5.3. Location of Complex Roots 181
(b) Let b, 1 b,-l >_... >_ bo > 0. Show that every zero w of the
polynomial g(t) must satisfy IwI < 1.
(c) Let ei be all positive and let 21 be the minimum and v the maxi-
mum of the quantities a,-l/a,, u,,-z/a,+~,... ,al/az, as/ar.
Show that every zero w of the polynomial f(t) must satisfy
u < lw] < 21.
(d) Verify that all the zeros w of 7t4 + 8t3 + 2t2 + 3t + 1 satisfy
l/4 < lull < 3/2.- Use the exercises to obtain estimates on the absolute values of the
zeros of the following polynomials. Compare the sharpness of the
different techniques.
(a) t2 -t + 1
(b) t5 - t4 - t3 + 4t2 - t - 1
(c) tll + ts - 3t5 + t4 + t3 - 29 + t - 2
(d) 3t2 - t + 4.- Schur-Cohn Criterion.
(a) Show that both zeros w of the real polynomial t2 + bt +c satisfy
1~1 < 1 if and only if lb1 < 1 + c < 2.
(b) Show that all the zeros w of the real polynomial t3 + bt2 + ct + d
satisfy 1~1 < 1 if and only ifIbd-cl < l-d2 Ib+dl < Il+cl.- A polynomial over the complex field is said to be stable if and only if
every one of its zeros has a negative real part. (The terminology arises
from applications; stability of a polynomial corresponds to physical
or biological stability of some system giving rise to the polynomial.)
(a) Show that a real stable polynomial must be the product of a
real constant and factors of the form t + r and t2 + bt + c, where
r, b and c are positive.
(b) Show that the signs of all the coefficients of a real stable poly-
nomial must be the same.
(c) Show that a linear or quadratic real polynomial is stable if and
only if all the coefficients are of the same sign.
(d) Give an example of a cubic polynomial whose coefficients are all
positive, but which is not stable.- We obtain a criterion for stability of the cubic polynomial
f(t) = at3 + bt2 + ct + d (a > 0).