196 6. Symmetric Functions of the Zeros(b) Let (Y = a - b - c,P=b-c-u,y=c-a-b,S=a+b+c.
Verify that
a+P+y+6=0
(YP+CYY+a6+py+Pb+y6=(a+P)(r+a)+(YP+YS
= -2(u2 + b2 + c2)cr& + C@ + ayb + p-16 = (a + P)rb + aP(r + 5) = 8abc
a,f?yb = (u2 + b2 + c2)2 - 4(u2b2 + u2c2 + b2c2).(c) Make use of (2) to show that u2, b2, c2 are the roots of the
equation 64t3 + 32pt2 + 4(p2 - 4r)t - q2 = 0.
(d) From (b) and (c), show that Q, p, y, 6 are the roots of (1).6.2 The Discriminant
In the theory of the quadratic polynomial ut2+ bt+c, it is possible to detect
the presence of a double zero by examining the discriminant b2 - 4ac,
a function of the coefficients. For the cubic polynomial t3 + pt + q, the
expression 27q2 + 4p3 plays a similar role. When the coefficients are real,
the signs of these expressions determine whether all the zeros are real.
(See Exercises 1.2.1, 1.4.4.) In Exercise 1.9, it was found that the zeros of
t3 + at2 + bt + c were real if and only if a2b2 ‘- 4u3c + 18abc - 4b3 - 27c2
was nonnegative.
For a polynomial of arbitrary degree, a function of the coefficients can be
determined which will vanish precisely when the polynomial has a nonsim-
ple zero and, for real polynomials, will be nonnegative if (but not necessar-
ily, only if) all zeros are real. The link between zeros and coefficients will
be Gauss’ theorem for symmetric functions. (See Exercises 1.5.10, 2.2.15.)
Exercises
- Let p(t) = ant” +... + a0 be a polynomial with zeros tl,... , t,.
(a) Consider the expression(t1 - tz)(t1 - t3)... (t2 - t3)(t2 - t4) * * * (t3 - t4) * * f (&-I - &),which is a product of
n
( >2 terms (one for each pair of zeros).
Verify that it will vanish exactly when there is a zero of multi-
plicity exceeding 1, but that it is not symmetric in the zeros.