Unknown

6

Symmetric Functions of the

Zeros

6.1 Interpreting the Coefficients of a Polynomial

Does the polynomial t6+2t5+3t4 -4t3+5t2+6t+7 have any nonreal zeros?

Some methods for answering this question have already been discussed,

but there is another approach which exploits the relationship between the

coefficients and the roots. If the zeros tl, t2,... , ts were real, then

Q = tf + t; +.. * + 1;

would have to be positive. Is there a way of determining Q without having
to go to all the trouble of finding the zeros?
& is a symmetric function of the zeros. In Exercise 4.6.6, it was indicated
that, as a result of the Fundamental Theorem of Algebra, the elementary
symmetric functions of the zeros are expressible in terms of the coefficients.
In Exercise 2.2.15, Gauss’ Theorem that every symmetric polynomial can
be written in terms of the elementary symmetric functions was presented.
As a consequence, we can obtain an expression for Q in terms of the coef-
ficients and, thus, check its sign. If Q turns out to be negative, we can be
sure that not all the zeros are real. If Q turns out to be nonnegative, then
there may or may not be nonreal zeros.
Already, for polynomials of low degree, we have exploited the relationship
between zeros and coefficients (see, for example, Exercise 1.2.16 and 4.1.7-
9). With the Fundamental Theorem in hand, we can generalize this to
polynomials of arbitrary degree.
Recall that, if ti (1 5 i < n) are the zeros of a,t” + +.. + alt + ac and s,
is the rth elementary symmetric function of the ti, then

sr = (-l)‘a,-,/a,.

Exercises

1. Find a cubic equation whose roots are the squares of the roots of the
   equation z3 - x2 + 3x - 10 = 0.