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180 5. Approximation and Location of Zeros

Exercises

1. Let w be any zero of the polynomial a,t” + a,-lt”-’ +.... Show that

(a) w = o;l[-a,-i - fzn-2wB1 -I. f - ai~-“+~ - a~w~~+l]

@I lwl 5 bnl-l c;=o bkl-

2. Let fc,~i,rs ,..., 2, be complex numbers. Show that

(a) IZI +z2++..+znl 6 IzII+I~~I+-~~+I~~I

(b) (zo+rl +-+.+~,a( 2 Izol- Ial- Izzl-...- Iznl.

When does equality occur?

3. Cauchy’s estimate. Let p(t) = a,t” + ... + ac and let K be the
   maximum of the absolute values of as/a,, al/a,,... ,a,-i/a,. Show
   that, if w is a zero of p(t), then

(a) 0 = a,w”[l+(a,~~/a,)w~‘+~~~+(a~/a,)w~”+’+(ao/a,)w~”]

(b) lwl < K + 1.

4. (a) Show that, if w is a nonvanishing zero of the polynomial p(t),
   then 20-l is a root of the polynomial t”p(t-‘).
   (b) Use (a) and th e result of Exercise 3 to determine a lower bound
   for the absolute value of a nonzero root of a polynomial equation.


5. (a) Let n _> 1, and suppose ai (0 5 i 5 n) are complex numbers
   with a0 # 0. Show that the polynomial

t” - Jan&-l - lan-21tn-2 -... - lqlt - laoI

has a unique positive zero r.

(b) Prove that every zero of the polynomial

t” + Q,-lP--l +. * * + qt + al-J

satisfies ]w] 5 r, where r is defined as in (a).

6. The polynomial at + b has the zero -b/a. In (c), this will be general-
   ized to an estimate for the zeros of polynomials of higher degree. In
   what follows, n 1 2,

g(t) = b,t” +... + b,t + b.

f(t) = a,P +.. * + a1t + ao.

(a) Let 0 < b, 5 bnwl 5 bn-2 _< e-e < bl 5 bo. Show that every

zero w of the polynomial g(t) must satisfy ]w] > 1.