194 6. Symmetric Functions of the Zeros
- Let ZL, v, w be the zeros of the cubic polynomial 4t3 - 7t2 - 3t + 2.
Determine a cubic polynomial whose zeros are u-(~/VW), r~-(l/wu),
w - (l/uv). - Let m, n,p, q be the zeros of the quartic polynomial t4-3t3+2t2+t-1.
Without determining any of m, n, p, q explicitly, determine a sextic
polynomial whose zeros are mn, mp, mq, np, nq, pq. Check your
answer in an independent way. - Find the manic polynomial whose zeros are the reciprocals of those of
the polynomial t3 - 2t2 + 6t + 5. Find the polynomial of degree 3 over
Z with these zeros for which the coefficients have greatest common
divisor 1. - Let p(t) = Ca,tP be a polynomial over C. Using the relationship
between zeros and coefficients, verify that a polynomial whose zeros
are the reciprocals of those of p(t) is
a# + a#-’ + a&-2 + * .. + a,-It + a, = Pp(l/t).- Let p(t) = Ca,.P have zeros ti. Determine a polynomial whose zeros
are kti (1 5 i 5 n). - (a) The sum of the zeros (counting multiplicity) of a polynomial is
0. Prove that the sum of the zeros of the derivative is also 0.
(b) If%1 ,... , zn are the zeros of a polynomial p(t) = tn +a,,-#‘-2 +
;,*:;;gy.., wn-l are the zeros of p’(t), prove that ncwf =- (a) Show that, if all the zeros of a polynomial p(t) = Cart’ are real,
then azsl 1 2a,-2a, an d a: 2 2aoa2. Show that the converse
is not true.
(b) Use (a) to verify that not all the zeros of the polynomial t6 +
2t5 + 3t4 - 4t3 + 5t2 + 6t + 7 are real.
(c) By making use of Rolle’s Theorem, strengthen the result of (a)
to: if the zeros are real, then (n - 1)~;~, 2 2na,a,-2. Give an
example to show that the converse is not true for n 2 3. - Consider the cubic polynomial p(t) = t3 + at2 + bt + c, with a, b, c
real. Suppose its zeros are x, y, z.
(a) Verify that xy = ,r2 + az + b and that(x - Y)~ = --[3z2 + 2az - (a” - 4b)] = (a” - 3b) -p’(z).