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Vieta's formulas applied to quadratic and cubic polynomials: The roots ... Note that the roots of the polynomial in the ... of the powers from the sum of the roots ...
This can be proved as follows. First, if r is a root of a polynomial with real coefficients, then its complex conjugate is also a root. So the non-real roots, if any, occur as pairs of complex conjugate roots. As a cubic polynomial has three roots (not necessarily distinct) by the fundamental theorem of algebra, at least one root must be real.
The Fermat cubic, in which the sum of three cubes equals another cube, has a general solution. The power sum symmetric polynomial is a building block for symmetric polynomials. The sum of the reciprocals of all perfect powers including duplicates (but not including 1) equals 1. The Erdős–Moser equation, + + + = (+) where m and k are positive ...
The roots, stationary points, inflection point and concavity of a cubic polynomial x 3 − 6x 2 + 9x − 4 (solid black curve) and its first (dashed red) and second (dotted orange) derivatives. The critical points of a cubic function are its stationary points , that is the points where the slope of the function is zero. [ 2 ]
The sum of a root and its conjugate is twice its real part. These three sums are the three real roots of the cubic polynomial +, and the primitive seventh roots of unity are , where r runs over the roots of the above polynomial. As for every cubic polynomial, these roots may be expressed in terms of square and cube roots.
The Gauss–Lucas theorem states that the convex hull of the roots of a polynomial contains the roots of the derivative of the polynomial. A sometimes useful corollary is that, if all roots of a polynomial have positive real part, then so do the roots of all derivatives of the polynomial. A related result is Bernstein's inequality.
A cubic polynomial has three zeroes in the complex number plane, which in general form a triangle, and the Gauss–Lucas theorem states that the roots of its derivative lie within this triangle. Marden's theorem states their location within this triangle more precisely:
These two roots are also a root of the derivative of the polyomial. So, they are also a root of the greatest common divisor of the polynomial and its derivative, which can be computed with the Euclidean algorithm for polynomials. It follows that the three roots are real, and if the coefficients are rational numbers, the same is true for the roots.