In algebra, * casus irreducibilis* (Latin for “the irreducible case”) is one of the cases that may arise in solving polynomials of degree 3 or higher with integer coefficients algebraically (as opposed to numerically), i.e., by obtaining roots that are expressed with radicals. It shows that many algebraic numbers are real-valued but cannot be expressed in radicals without introducing complex numbers. The most notable occurrence of

*casus irreducibilis*is in the case of cubic polynomials that have three real roots, which was proven by Pierre Wantzel in 1843.[1] One can see whether a given cubic polynomial is in so-called

*casus irreducibilis*by looking at the discriminant, via Cardano’s formula.[2]

## . . . Casus irreducibilis . . .

Let

be a cubic equation with

${displaystyle aneq 0}$. Then the discriminant is given by

It appears in the algebraic solution and is the square of the product

of the

${displaystyle {tbinom {3}{2}}=3}$differences of the 3 roots

${displaystyle x_{1},x_{2},x_{3}}$.[3]

- If
*D*, then the polynomial has one real root and two complex non-real roots. - If
*D*= 0, then - If
*D*> 0, then

## . . . Casus irreducibilis . . .

*
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