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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

$a_0{x}^n+a_1{x}^{n-1}+\cdots +a_{n-1}x+a_n=0$

and if $b_0,b_1,\dots ,b_{n-1},b_n$ are numbers in arithmetic progression, then r is also a root of

$a_0{b}_0{x}^n+a_1{b}_1{x}^{n-1}+\cdots +a_{n-1}{b}_{n-1}x+a_n{b}_n=0.$

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

$a_0{x}^n+a_1{x}^{n-1}+\cdots +a_{n-1}x+a_n$

takes on a relative maximum or minimum value, then a is a root of the equation

$n{a}_0{x}^n+(n-1)a_1{x}^{n-1}+\cdots +2a_{n-2}{x}^2+a_{n-1}x=0$

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.

See also

• Johann Hudde