Cubic equation

In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. An example is the equation

The nature of the roots

Every cubic equation with real coefficients has at least one solution x among the real numbers; this is a consequence of the intermediate value theorem. We can distinguish several possible cases using the discriminant,

Related Topics:
Real - Intermediate value theorem - Discriminant

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: Delta = 4lpha_1^3lpha_3 - lpha_1^2lpha_2^2 + 4lpha_0lpha_2^3 - 18lpha_0lpha_1lpha_2lpha_3 + 27lpha_0^2lpha_3^2.

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The following cases need to be considered.

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• If Δ < 0, then the equation has three distinct real roots.
• If Δ > 0, then the equation has one real root and a pair of complex conjugate roots.
• If Δ = 0, then (at least) two roots coincide. To decide how many distinct roots there are, we define

:: Delta_2 = 2lpha_2^3 - 9lpha_1lpha_2lpha_3 + 27lpha_0lpha_3^2,

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: and consider two further cases.

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:* If Δ2 = 0, then all three roots coincide and we have a triple real root.

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:* Otherwise, the equation has a double real root and a single real root.

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:The number Δ2 is the resultant of the cubic and its second derivative.

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: See also: multiplicity of a root of a polynomial

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