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It has been seen in § Low degrees that the sign of the discriminant provides useful information on the nature of the roots for polynomials of degree 2 and 3. For higher degrees, the information provided by the discriminant is less complete, but still useful. More precisely, for a polynomial of degree n, one has:
The discriminant of a polynomial is a function of its coefficients that is zero if and only if the polynomial has a multiple root, or, if it is divisible by the square of a non-constant polynomial. In other words, the discriminant is nonzero if and only if the polynomial is square-free.
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 ]
Repeated discriminants: the discriminant of a quadratic field uniquely identifies it, but this is not true, in general, for higher-degree number fields. For example, there are two non-isomorphic cubic fields of discriminant 3969. They are obtained by adjoining a root of the polynomial x 3 − 21x + 28 or x 3 − 21x − 35, respectively. [7]
It has the smallest discriminant of all totally real cubic fields, namely 49. [4] The field obtained by adjoining to Q a root of x 3 + x 2 − 3x − 1 is an example of a totally real cubic field that is not cyclic. Its discriminant is 148, the smallest discriminant of a non-cyclic totally real cubic field. [5]
The algebra of invariants of a ternary cubic under SL 3 (C) is a polynomial algebra generated by two invariants S and T of degrees 4 and 6, called Aronhold invariants. The invariants are rather complicated when written as polynomials in the coefficients of the ternary cubic, and are given explicitly in (Sturmfels 1993, 4.4.7, 4.5.3)
Rearranging gives 8x 3 − 6x − 1 = 0, which fails the rational root test as none of the rational numbers suggested by the theorem is actually a root. Therefore, the minimal polynomial of cos(20°) has degree 3, whereas the degree of the minimal polynomial of any constructible number must be a power of two. Expressing cos(20°) in radicals ...
Degree: The maximum exponents among the monomials. Factor: An expression being multiplied. Linear factor: A factor of degree one. Coefficient: An expression multiplying one of the monomials of the polynomial. Root (or zero) of a polynomial: Given a polynomial p(x), the x values that satisfy p(x) = 0 are called roots (or zeroes) of the polynomial p.