# Hudde's rules

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

${\displaystyle a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}=0\,}$
and if ${\displaystyle b_{0},b_{1},\dots ,b_{n-1},b_{n}}$ are numbers in arithmetic progression, then r is also a root of
${\displaystyle a_{0}b_{0}x^{n}+a_{1}b_{1}x^{n-1}+\cdots +a_{n-1}b_{n-1}x+a_{n}b_{n}=0.\,}$
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

${\displaystyle a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}\,}$
takes on a relative maximum or minimum value, then a is a root of the equation
${\displaystyle na_{0}x^{n}+(n-1)a_{1}x^{n-1}+\cdots +2a_{n-2}x^{2}+a_{n-1}x=0.\,}$
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.

## References

• Carl B. Boyer, A history of mathematics, 2nd edition, by John Wiley & Sons, Inc., page 373, 1991.