Search for a tool
Discriminant of a Polynomial

Tool to compute the discriminant of a polynomial to deduce its roots (values or the expression is zero, equal to 0).

Results

Discriminant of a Polynomial -

Tag(s) : Functions

Share
Share
dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!


Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!


Feedback and suggestions are welcome so that dCode offers the best 'Discriminant of a Polynomial' tool for free! Thank you!

Discriminant of a Polynomial

Discriminant Calculator



Root Calculator

⮞ Go to: Polynomial Root

Answers to Questions (FAQ)

What is a discriminant? (Definition)

A discriminant of a polynomial is an expression giving information about the number and the nature of the roots of the polynomial.

How to calculate a discriminant?

For a quadratic polynomial $ ax^2+bx+c $, the discriminant named delta $ \Delta $ is calculated with the formula:

$$ \Delta = b^2-4ac $$

The fact of knowing the value of the discriminant then solves the equation more easily through formulas (using this discriminant).

Example: The equation $ 2x^2+3x+1 = 0 $ of type $ ax^2+bx+c $ (with $ a = 2 $, $ b = 3 $ et $ c = 1 $) has for discriminant $ \Delta = b^2-4ac = 3^2-4*2*1 = 1 $

For a cubic polynomial of the form $ ax^3+bx^2+cx+d $ the discriminant formula is

$$ \Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd $$

For a polynomial of degree 1 or 0 the discriminant is not generally calculated, its value has no interest, it is nevertheless sometimes defined at the value $ 1 $.

What is the link between the discriminant and the roots of a polynomial?

The discriminant acts as an indicator of whether the polynomial has real, distinct or imaginary roots and their number.

For a polynomial of degree 2 if the discriminant is equal to $ 0 $, then the polynomial has a unique (double) root. If the discriminant is positive, then it has 2 real roots, and if it is negative, it has 2 complex and conjugate roots.

How to find the roots of a polynomial with the discriminant?

For a quadratic polynomial of type $ ax^2+bx+c = 0 $

If the discriminant is positive (strictly), the equation has two solutions x1 and x2:

$$ x_1 = \frac {-b + \sqrt \Delta}{2a} \\ x_2 = \frac {-b - \sqrt \Delta}{2a} $$

Example: The equation $ 2x^2+3x+1 = 0 $ has for discriminant $ \Delta = 1 $, so solutions are $ x_1 = -1/2 $ and $ x_2 = -1 $

If the discriminant is zero, the equation has a double root:

$$ x_1=x_2 = -\frac b{2a} $$

If the discriminant is negative (strictly), the equation has 2 complex conjugate solutions:

$$ \delta^2 = \Delta $$

$$ x_1 = \frac {-b + \delta}{2a} \\ x_2 = \frac {-b - \delta}{2a} $$

For equations of higher degrees (degree 3 or 4 or more), knowledge of the discriminant allows us to know the number of roots, however there is no formula allowing them to be found from the discriminant.

Source code

dCode retains ownership of the "Discriminant of a Polynomial" source code. Except explicit open source licence (indicated Creative Commons / free), the "Discriminant of a Polynomial" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Discriminant of a Polynomial" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Discriminant of a Polynomial" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.

Cite dCode

The copy-paste of the page "Discriminant of a Polynomial" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Discriminant of a Polynomial on dCode.fr [online website], retrieved on 2024-06-16, https://www.dcode.fr/polynomial-discriminant

Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Questions / Comments

Feedback and suggestions are welcome so that dCode offers the best 'Discriminant of a Polynomial' tool for free! Thank you!


https://www.dcode.fr/polynomial-discriminant
© 2024 dCode — El 'kit de herramientas' definitivo para resolver todos los juegos/acertijos/geocaching/CTF.
 
Feedback