182 CHAPTER 3 Inequalities
3.7.3 A special class of trinomials
We shall start this discussion with a specific example. Let f(x) =
a 3 x^3 +a 2 x^2 +a 1 x+a 0 and let g(x) = b 2 x^2 +b 1 x+b 0 and form the
Sylvester matrix
S(f,g) =
a 3 a 2 a 1 a 0 0
0 a 3 a 2 a 1 a 0
b 2 b 1 b 0 0 0
0 b 2 b 1 b 0 0
0 0 b 2 b 1 b 0
.
Next assume that in the above, we actually havea 3 =a 2 = 0, and that
b 26 = 0. Then the determinant of the above is given by
detS(f,g) = det
b 2 0 0 0 0
0 b 2 0 0 0
0 0 a 1 a 0 0
0 0 0 a 1 a 0
0 0 b 2 b 1 b 0
=b^22 R(a 1 x+a 0 ,b 2 x^2 +b 1 x+b 0 ).In general, assume that
f(x) =anxn+an− 1 xn−^1 +···+a 1 x+a 0and that
g(x) =bmxm+bm− 1 xm−^1 +···+b 1 x+b 0 , bm 6 = 0.If we have ak 6 = 0 and that ak+1 = ak+2 = ··· = an = 0, then one
patterns an argument based on the above to arrive at the conclusion:
Lemma 3. With hypotheses as above,
detS(f,g) = (−1)m(n−k)bnm−kR(f,g).Assume now that we are given polynomials f(x) = anxn+ lower,
andg(x) =bmxm+ lower. The Sylvester matrixS(f,g) has in rows 1
through mthe coefficients of f(x) and in rowsm+ 1 through m+n