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5.4. Problems 187

11. Prove that the roots of the cubic equation

t3 - (a + b + c)t2 + (ab + bc + co - d2 - e2 - f2)t

+ (ad2 + be2 + cf2 - abc - 2def) = 0

are real when a, b, c, d, e, f are real.

12. Determine each real root of

x4 - (2.1OlO + 1) x2 - 2 + 1020 + 1o’O - 1 = 0

correct to four decimal places (a denotes multiplication).

13. Consider polynomials .x2 - bx+c with integer coefficients which have
    two distinct zeros in the open interval (0,l). Exhibit with a proof the
    least positive integer value of a for which such a polynomial exists.


14. The sequence {qn(x)} of polynomials is defined by

91(x) = 1 +x 42(x) = 1+ 2x

and, for m 1 1 by

Q2m+l(X) = q2m(X) + (m + l)XqZm-l(X)

92m+2(X) = Q2m+l(X) + (m + l)XQ2m(X).

Let x,, be the largest real solution of qn(x) = 0. Prove that {x,,} is

an increasing sequence whose limit is 0.

15. Assuming that all the zeros of the cubic t3 + at2 + bt + c are real,
    show that the difference between the greatest and the least of them
    is not less than (u” - 3b)‘j2 nor greater than 2(a2 - 3b)‘i2.


16. How many roots of the equation z6 + 6% + 10 = 0 lie in each quadrant
    of the complex plane?


17. Show that

4x172 + 1)64 - 3X9(X + 1)27 + 224(x + 1)” - 1 = 0

has at most 14 positive roots.

18. For which real values of a do all roots of z3 - z2 + a = 0 satisfy
    I%1 2 l?


19. Let the zeros a, b, c of f (t) = t3+pt2+qt+r be real, and let a 2 b 5 c.
    Prove that, if the interval (b, c) is divided into six equal parts, a zero
    of f’(x) will be found in the fourth part, counting from the end 6.
    What will be the form of f(t) if the root in question of f’(t) = 0 falls
    at either end of the fourth part?