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In mathematics, a cubic function is a function of the form () = + + +, that is, a polynomial function of degree three. In many texts, the coefficients a , b , c , and d are supposed to be real numbers , and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to ...
The eigenvalues of a 3×3 matrix are the roots of a cubic polynomial which is the characteristic polynomial of the matrix. The characteristic equation of a third-order constant coefficients or Cauchy–Euler (equidimensional variable coefficients) linear differential equation or difference equation is a cubic equation.
The polynomial P(x) has a rational root (this can be determined using the rational root theorem). The resolvent cubic R 3 (y) has a root of the form α 2, for some non-null rational number α (again, this can be determined using the rational root theorem). The number a 2 2 − 4a 0 is the square of a rational number and a 1 = 0. Indeed:
Similarly, the discriminant of a cubic polynomial is zero if and only if the polynomial has a multiple root. In the case of a cubic with real coefficients, the discriminant is positive if the polynomial has three distinct real roots, and negative if it has one real root and two distinct complex conjugate roots.
If the polynomial to be factored is + + + +, then all possible linear factors are of the form , where is an integer factor of and is an integer factor of . All possible combinations of integer factors can be tested for validity, and each valid one can be factored out using polynomial long division .
Since every polynomial with complex coefficients can be factored into 1st-degree factors (that is one way of stating the fundamental theorem of algebra), it follows that every polynomial with real coefficients can be factored into factors of degree no higher than 2: just 1st-degree and quadratic factors.
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