188 5. Approximation and Location of Zeros
- Let 731 < 782 < ... < nk be a set of positive integers. Prove that the
polynomial 1 + z”l + zn3 +... + 9’~ has no zeros inside the circle
l-4 < (1/2)(&- 1). - For which complex values of a do all the zeros of z3 + 12( 1 + ifi)% + a
lie on a straight line? - Show that 1 + t + t2/2! + t3/3! +... + t2”/(2n)! = 0 has no real roots.
- Prove that a polynomial p(t) f or which p(t) is real when t is real and
nonreal when t is nonreal must be linear. - Let p(t) = ao+alt+.. .+a”t” be a real polynomial of degree exceeding
1, such that
I”/21
0 < a0 < - x[1/(2k+ l)]a2k.
k=l
Prove that p(t) h as a real zero r such that Irl < 1. - p(z) is a complex polynomial whose zeros can be covered by a closed
circular disc of radius R. Show that the zeros of rip(z)) - kp’(z) can
be covered by a closed circular disc of radius R + lkl, where n is the
degree of p(z), k is any complex number and p’(z) is the derivative
of p(z). - Find all the zeros of azPt* - bzp + b - a (0 < a < b) which satisfy
I%[ = 1. - Suppose that -1 < u 5 1. Prove that each root of the equation
has modulus 1.2 “i-1 - ux” f ‘212 - 1 = 0- Show that, for all integers k >_ 0,
tn + 1) -kx” + n-Lx+1 + * *. + 2-kx + 1 = 0has no real root if n is even and exactly one root if n is odd.- Let al, 02,... , a” be nonzero reals with al < 02 <... < a”. Show
that the following equation holds for n real values of x:
-+-^01 02 +...+%==,
al-x 02 -x 0” - x
if all the oi have the same sign. What happens if al < 0 < a,?- Let k > 0. Show that, if lail < k (1 5 i 5 n), then
1+ 01z + 0222 +... + 0”Z” = 0
has no root with IzI < l/(k + 1). Is the converse true?