In mathematics, Hudde's rules are two properties of polynomial roots described by Johann Hudde.
1. If r is a double root of the polynomial equation
and if
b
0
,
b
1
,
…
,
b
n
−
1
,
b
n
are numbers in arithmetic progression, then
r is also a root of
a
0
b
0
x
n
+
a
1
b
1
x
n
−
1
+
⋯
+
a
n
−
1
b
n
−
1
x
+
a
n
b
n
=
0.
This definition is a form of the modern theorem that if
r is a double root of
ƒ(
x) = 0, then
r is a root of
ƒ '(
x) = 0.
2. If for x = a the polynomial
takes on a relative maximum or minimum value, then
a is a root of the equation
n
a
0
x
n
+
(
n
−
1
)
a
1
x
n
−
1
+
⋯
+
2
a
n
−
2
x
2
+
a
n
−
1
x
=
0.
This definition is a modification of Fermat's theorem in the form that if
ƒ(
a) is a relative maximum or minimum value of a polynomial
ƒ(
x), then
ƒ '(
a) = 0.
