202 6. Symmetric Functions of the Zeros
- The product of two of the four roots of the quartic equation
x4 - 18x3 + kx2 + 200x - 1984 = 0is -32. Determine k.- Three of the roots of x4 -px3 + qx2 - rx + s = 0 are tan A, tan B and
tanC, where A, B, C are angles of a triangle. Determine the fourth
root as a function of p, q, r, s alone. - If x, y, z are real and satisfy x + y + z = 5 and yr + 2x + xy = 3,
prove that -1 5 z 5 13/3. - Let u, V, w be the roots of x3 - 6x2 + ax + a = 0. Determine all real
a for which (U - 1)3 + (V - 2)3 + (w - 3)3 = 0. For each a determine
the corresponding values of U, v, 20. - Determine those values of the real number a and positive integer n
exceeding 1 for which
” xk+2
c k=l --n-3 xk - ’where x1,22,..., xn are the zeros of x” + ax”-’ + on-lx + 1.- Ifu+v+w=O,provethat,forn=0,1,2 ,...,
u”+3 + ?l”+3 + w”+3 = u?lw(u” + v” + w”)+ (1/2)(U2 + V2 + Wz)(u”+l + Z)*+l + w*+l)*- Let p, q, r, s be the roots of the quartic equation x4-ax3+bx2-cx+d.
Find the quartic whose roots are pq + qr + rp, pq + qs + sp, pr+ rs + sp,
qr + rs + sq. - Find all integer values of a such that all the zeros of t4 - 14t3 + 61t2 -
84t + a are integers. - Suppose that p( I ) is a polynomial of degree at least 2 whose coeffi-
cients are complex numbers and not all real. Prove that the equation
PWPW = ~(4 h as roots in both the upper and lower half planes
(i.e. for which Im,r take both positive and negative values). Give an
example to show that this is false when the degree of p(z) is 1. - Determine all polynomials of degree n with each of its n+l coefficients
equal to +l or -1 which have only real zeros. - Suppose that every zero of the polynomial f(x) is simple. If every
root of the equation f’(x) = 0 be subtracted from every root of the
equation f(x) = 0, find the sum of the reciprocals of the differences.