6.2. The Discriminant 197(b) Let D(tl,h ,... , tn) be the square of the expression in (a). This
is called the discriminant of the polynomial p(t). Verify that
it is a homogeneous symmetric polynomial of degree n(n - 1)
and that it vanishes exactly when there is a zero of multiplicity
greater than 1.
(c) Must the discriminant of a polynomial over R be real?- Verify that for the quadratic polynomial at2 + bt +c, the discriminant,
as defined in Exercise l(b) is (b/u)2 - 4(c/a) (which is equal to the
usual discriminant divided by a square). - (a) Suppose that p(t) is a polynomial over R with discriminant D
whose zeros are all real. Prove that D 10.
(b) Show that the converse of (a) is true for quadratic and cubic
polynomials, but not for polynomials of higher degree. - Show that if p(t) is a polynomial over R with all zeros distinct, then
(i) if there are an odd number of pairs of nonreal complex conju-
gates among the zeros, then D < 0;
(ii) if there are an even number of pairs of nonreal complex conju-
gates among the zeros, then D > 0.- (a) Find the discriminant of the polynomial t3 +pt + q and state its
relationship to the quantity 27q2 + 4p3.
(b) Find the discriminant of the cubic t3 + at2 + bt + c and of the
general cubic azt3 + a2t2 + alt + ao. - Find the zeros of each of the following quartics and use them to
evaluate their discriminants:
(a) t4 - 1
(b) t4 + 5t2 + 4.- Show that the discriminant of a polynomial over C is nonzero if and
only if the greatest common divisor of the polynomial and its deriva-
tive is a nonzero constant.
Exploration
E.51. What is the discriminant oft” - l?