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.
