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Discriminant of a Polynomial

Tool to compute the discriminant of a polynomial. A discriminant of a polynomial is an expression giving information about the nature of the roots of the polynomial.

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Discriminant of a Polynomial -

Tag(s) : Arithmetics, Mathematics

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Discriminant of a Polynomial

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Tool to compute the discriminant of a polynomial. A discriminant of a polynomial is an expression giving information about the nature of the roots of the polynomial.

Answers to Questions

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 \( \Delta = b^2-4ac = 3^2-4*2*1 = 1 \)

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

For a quadratic polynomial of type \( ax^2+bx+c = 0 \)

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

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

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

Example: The equation \( 2x^2+3x+1 = 0 \) has \( \Delta = 1 \) 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 complex solutions:

$$ \delta^2 = \Delta $$

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

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

For equations of higher degrees, the calculations are much more complicated, but knowledge of the determinants is important.

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