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

Factorization

If r is any root of (1), then we may factor using r to obtain

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:(x-r)(x^2+(a+r)x+b+ar+r^2) = x^3+ax^2+bx+c.

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Hence if we know one root we can find the other two by solving a quadratic equation, giving

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: rac12 left(-a-r pm sqrt{-3r^2-2ar+a^2-4b} ight)

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for the other two roots. If we are finding the roots of a polynomial with real coefficients and one real root, we can find the real root purely in terms of the real (rather than complex) cube root function, or alternatively stated we can find the root by extracting cube roots only of positive quantities. The complex conjugate roots can then be found as above.

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