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

Discriminant of a Polynomial - dCode

Tag(s) : Functions

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

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 $.

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.

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.

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*Discriminant of a Polynomial* on dCode.fr [online website], retrieved on 2024-06-16,

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