Advanced High-School Mathematics

176 CHAPTER 3 Inequalities

From the above, we see that

∆ = ∆(f) =a^2 nn−^2

∏

1 ≤i<j≤n

(xj−xi)^2.

The difficulty with the above expression is that its computation ap-
pears to require the zeros off(x). However, this is a symmetric poly-
nomial in the “variables”x 1 , x 2 , ..., xn and hence^7 can be written as
a polynomial in theelementary symmetric polynomials

σ 1 =x 1 +x 2 +···+σn

σ 2 =x 1 x 2 +x 1 x 3 +···=

∑

i<j

xixj

σ 3 =

∑

i<j<k

xixjxk

σn=x 1 x 2 ···xn

This was carried out for the quadratic polynomialf(x) =ax^2 +bx+con
page 168; the result for the cubic polynomialf(x) =ax^3 +bx^2 +cx+d
was given on page 171. Carrying this out for higher degree polynomials
is quite difficult in practice (see, e.g., Exercise 5, above). The next
two subsections will lead to a more direct (but still computationally
very complex) method for computing the discriminant of an arbitrary
polynomial.

3.7.1 The resultant off(x) andg(x)

Let f(x) = anxn+an− 1 xn−^1 +···+a 1 x+a 0 , and g(x) = bmxm+
bm− 1 xm−^1 +···+b 1 x+b 0 be polynomials having real coefficients of
degreesnandm, respectively. Define the (n+m)×(n+m)Sylvester
matrixrelative tof(x) andg(x),S(f,g) by setting

(^7) This follows from the so-calledFundamental Theorem on Symmetric Polynomials.