Definition:Irreducible Case of Cubic

Definition

Let $P$ be the cubic equation:

$a x^3 + b x^2 + c x + d = 0$

such that $a, b, c, d \in \R$ with $a \ne 0$.

$P$ is an instance of the irreducible case if and only if the discriminant is less than $0$.

That is, when $P$ has $3$ real unequal roots.

Also see

• Cardano's Formula for Real Coefficients

• Results about cubic equations can be found here.

Historical Note

The irreducible case of the cubic was the one which caused difficulty in Cardano's Formula for its solution.

This is because it leads to manipulations that require the cube root of an imaginary number.

When this area of algebra was initiated, no such concept was understood.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cubic
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cubic