Question

What is the maximum number of real distinct roots that a quartic equation can have?

Answer 1

An nth degree polynomial can have as many as n real roots.

Answer 2

The number of distinct real roots
1
of the cubic equation
x
3
+
bx
2
+
cx
+
d
= 0 (1)
equals, for example, the number of normals to the parabola
y
=
x
2
through a
given point in the plane (Bains and Thoo (2007)),
2
as well as the number of
equilibrium solutions of
dx/dt
=
x
3
+
bx
2
+
cx
+
d
.
Now, one way to find the number of real roots of (1) is to solve th
e equation.
Certainly, (1) can be solved by hand easily if
d
= 0. If
d
6
= 0, the equation can
still be solved by hand using Cardan’s (or Cardano’s) cubic f
ormula (Fine (1961);
Gellert et al. (1975)),
3
but not easily in general. Of course, using computer
software or a graphing calculator can make light work of solv
ing (1) altogether.
However, there is a certain satisfaction in being able to tel
l the number of real
roots of (1) without first solving the equation. It turns out t
hat this is easy to
do. The key is the
discriminant
.
Every intermediate algebra student learns the quadratic fo
rmula, deriving
it by completing the square. And with the quadratic formula i
n hand, it is
apparent that the number of real roots of the quadratic equat
ion
x
2
+
bx
+
c
= 0
∗
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.
†
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.
1
We mean the number of
distinct
real roots throughout.
2
It is remarkable that Apollonius had obtained precise resul
ts on the number of normals to
a parabola through a given point using purely synthetic geom
etry (Heath, 1981, pp. 158–159,
163–166) almost 1700 years before Italian mathematicians s
olved the cubic.
3
For Cardan’s formulation of the solution of the cubic, as wel
l as the story behind the
quarrel between Cardan and Tartaglia over its publication,
see Burton (2003) for example.
For other formulations of the solution of the cubic, see Kalm
an and White (1998) and the
references therein.