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186 5. Approximation and Location of Zeros

5.4 Problems

1. Prove that, for complex k the polynomial t3 - 3t + k never has more
   than one real zero in the closed interval [0, 11.


2. Prove that the positive root of

x(x + 1)(x + 2)... (x + n) = 1

is less than l/n!.

3. Prove that all roots but one of the equation

nx” = ~+x+z~+..++x”-~

have absolute value less than 1.

4. Show that x4 - 5x3 - 4x2 - 7x + 4 = 0 has no negative roots.


5. Let fo(t) = t and f,,(t) = fn-l(t)2 - 2 for n 2 1. Show that the
   equation f,,(t) = 0 has 2” real roots.


6. Let a and b be unequal real numbers. Prove that

(a - b)t” + (a2 - b2)tn-1 +... + (a’- - b”+‘) = 0

has at most one real root.

7. Let al > Q2 > u3 > u4 > u5 > es, and let p = al +... + a6,
   4 = ala3 + (3305 + a5al + a$4 + a4a6 + a6a2, r = ala305 $- a2a4a6.
   Show that all the zeros of 2t3 - pt2 + qt - r are real.


8. Let a, b, c > 0. Show that the equation

x3 - (a” + b2 + c”)x - 2abc = 0

has a unique positive root u which satisfies

(2/3)(a + b + c) < u < a + b + c.

9. The equation (z - Q~)(x - ~2)... (x - a,) = 1 where Qi E R, has n
   real roots ri. Find the minimum number of real roots of the equation

(x - rl)(x - r2)... (x - rn) = -1.

10. Let p(x) be a polynomial of degree n with real roots al, as, a3,... , a,.
    Suppose the real number b satisfies

Prove that

lb - all < lb - ail (2 5 i < n).

IPWI 1 2 -“+llp’(a)(b - m)l.