Discriminant of a polynomial: Difference between revisions
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In [[algebra]], the '''discriminant of a polynomial''' is an invariant which determines whether or not a [[polynomial]] has repeated roots. | In [[algebra]], the '''discriminant of a polynomial''' is an invariant which determines whether or not a [[polynomial]] has repeated roots. | ||
Given a polynomial | Given a polynomial | ||
:<math>f(x)= a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 </math> | :<math>f(x)= a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 </math> | ||
with roots | with roots <math>\alpha_1,\ldots,\alpha_n </math>, the discriminant Δ(''f'') with respect to the variable ''x'' is defined as | ||
the discriminant Δ(''f'') with respect to the variable ''x'' is defined as | |||
:<math>\Delta = a_n^{2(n-1)} \prod_{i \neq j} (\alpha_i - \alpha_j) . </math> | :<math>\Delta = (-1)^{n(n-1)/2} a_n^{2(n-1)} \prod_{i \neq j} (\alpha_i - \alpha_j) . </math> | ||
The discriminant is thus zero if and only if ''f'' has a repeated root. | The discriminant is thus zero if and only if ''f'' has a repeated root. | ||
The discriminant may be obtained as the [[resultant (algebra)|resultant]] of the polynomial and its [[derivative]]. | The discriminant may be obtained as the [[resultant (algebra)|resultant]] of the polynomial and its [[formal derivative]]. | ||
==Examples== | ==Examples== | ||
The discriminant of a quadratic <math>aX^2 + bX + c</math> is <math>b^2 - 4ac</math>, which plays a key part in the solution of the [[quadratic equation]]. | The discriminant of a quadratic <math>aX^2 + bX + c</math> is <math>b^2 - 4ac</math>, which plays a key part in the solution of the [[quadratic equation]]. | ||
==References== | |||
* {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 | pages=193-194,204-205,325-326 }}[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 16:00, 7 August 2024
In algebra, the discriminant of a polynomial is an invariant which determines whether or not a polynomial has repeated roots.
Given a polynomial
with roots , the discriminant Δ(f) with respect to the variable x is defined as
The discriminant is thus zero if and only if f has a repeated root.
The discriminant may be obtained as the resultant of the polynomial and its formal derivative.
Examples
The discriminant of a quadratic is , which plays a key part in the solution of the quadratic equation.
References
- Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 193-194,204-205,325-326. ISBN 0-201-55540-9.