SECTION 3.7 Discriminant 183
it has the coefficients of g(x). Note that adding a multiple a of row
m+n to the firstmrows ofS(f,g) will produce the Sylvester matrix
S(f+ag,g), whose determinant is unchanged. Ifm < n, then adding
a times row m+n−1 to each of the first m rows of S(f,g) will
produce the Sylvester matrixS(f+axg,g) with the same determinant:
detS(f +axg,g)detS(f,g). More generally, we see that as long as
k≤n−m, then for any constanta, detS(f+axkg,g) = detS(f,g).
This easily implies the following very useful fact:
Theorem 6. Given the polynomials f(x), g(x) with real coefficients
of degreesn≥m(respectively), then for any polynomialh(x)of degree
≤n−m, detS(f+gh,g) =detS(f,g).
Now consider the monic trinomial of the form f(x) =xn+ax+b
wherea, bare real numbers. Applying Theorem 3, we see that
∆(f) = (−1)n(n−1)/^2 R(f,f′)
= (−1)n(n−1)/^2 detS(xn+ax+b,nxn−^1 +a)
= (−1)n(n−1)/^2 detS((a−a/n)x+b,nxn−^1 +a) (Theorem 3)
= (−1)n(n−1)/^2 (−1)(n−1)
2
nn−^1 R((a−a/n)x+b,nxn−^1 +a)
= (−1)n(n−1)/^2 (−1)(n−1)
2
nn−^1 (a−a/n)n−^1 (n(−1)n−^1 (b/(a−a/n))n−^1 +a)
(Corollary 3)
= (−1)n(n−1)/^2 (−1)n−^1 ((−1)n−^1 nnbn−^1 +an(n−1)n−^1 )
= (−1)n(n−1)/^2 (nnbn−^1 + (−1)n−^1 (n−1)n−^1 an).Example. Let f(x) =
3
x+x^3 , x > 0 and find the minimum valueoff(x). While such a problem is typically the province of differential
calculus, we can use a discriminant argument to obtain the solution.