Advanced High-School Mathematics

SECTION 3.6 Cubic Discriminant 171

∆<0 =⇒P(x) has one real zero and two non-real complex zeros.

This is all rounded out by the obvious statement that

∆ = 0 =⇒P(x) has a multiple zero and all zeros are real.

Of course, none of the above is worth much unless we have a method
of computing ∆. The trick is to proceed as in the quadratic case and
compute ∆ in terms of the coefficients of P(x). We start with the
observation that

P(x) =a(x−x 1 )(x−x 2 )(x−x 3 ),

all of which implies that (by expanding)

b=−a(x 1 +x 2 +x 3 ), c=a(x 1 x 2 +x 1 x 3 +x 2 x 3 ), d=−ax 1 x 2 x 3.

We set

σ 1 =x 1 +x 2 +x 3 , σ 2 =x 1 x 2 +x 1 x 3 +x 2 x 3 , σ 3 =x 1 x 2 x 3 ,

and call them theelementary symmetric polynomials(inx 1 , x 2 , x 3 ).
On the other hand, by expanding out ∆, one has that (after quite a bit
of very hard work!)

∆ = a^4 (x 3 −x 2 )^2 (x 3 −x 1 )^2 (x 2 −x 1 )^2

= a^4 (− 4 σ 13 σ 3 +σ^21 σ^22 + 18σ 1 σ 2 σ 3 − 4 σ 23 − 27 σ^23 )

= − 4 b^3 d+b^2 c^2 + 18abcd− 4 ac^3 − 27 a^2 d^2

giving a surprisingly complicated homogeneous polynomial in the co-
efficienta, b, c, andd. (See Exercise 6 below for a more more direct
method for computing ∆.)

We’ll close this section with a representative example. Keep in mind
that just as in the case of the quadratic, when the discriminant of a
cubic is 0, then the graph of this cubic is tangent to thex-axis at the
multiple zero.