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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 |
| | ::<math>a_0x^n + a_1x^{n-1} + \cdots + a_{n-1}x + a_n = 0 \, </math> |
| | :and if <math>b_0, b_1, \dots, b_{n-1}, b_n</math> are [[number]]s in [[arithmetic]] progression, then ''r'' is also a [[root]] of |
| | ::<math>a_0b_0x^n + a_1b_1x^{n-1} + \cdots + a_{n-1}b_{n-1}x + a_nb_n = 0. \, </math> |
| | :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''<nowiki> '</nowiki>(''x'') = 0. |
| | |
| | 2. If for ''x'' = ''a'' the polynomial |
| | ::<math>a_0x^n + a_1x^{n-1} + \cdots + a_{n-1}x + a_n \, </math> |
| | :takes on a relative [[maximum]] or [[minimum]] [[Value (mathematics)|value]], then a is a root of the equation |
| | ::<math>na_0x^n + (n-1)a_1x^{n-1} + \cdots + 2a_{n-2}x^2 + a_{n-1}x = 0 \, </math> |
| | :This definition is a modification of [[Fermat's theorem (stationary points)|Fermat's theorem]] in the form that if ''ƒ''(''a'') is a relative maximum or minimum value of a polynomial ''ƒ''(''x''), then ''ƒ''<nowiki> '</nowiki>(''a'') = 0. |
| | |
| | ==References== |
| | *Carl B. Boyer, A history of mathematics, 2nd edition, by John Wiley & Sons, Inc., page 373, 1991. |
| | |
| | ==See also== |
| | *[[Johann Hudde]] |
| | |
| | [[Category:Rules]] |
| | [[Category:Theorems in algebra]] |
| | [[Category:Polynomials]] |
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
are numbers in arithmetic progression, then r is also a root of
- 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
- 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