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

a_0x^n + a_1x^{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_0b_0x^n + a_1b_1x^{n-1} + \cdots + a_{n-1}b_{n-1}x + a_nb_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_0x^n + a_1x^{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

na_0x^n + (n-1)a_1x^{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.

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