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A cubic equation with real coefficients can be solved geometrically using compass, straightedge, and an angle trisector if and only if it has three real roots. [30]: Thm. 1 A cubic equation can be solved by compass-and-straightedge construction (without trisector) if and only if it has a rational root.
A cubic function with real coefficients has either one or three real roots (which may not be distinct); [1] all odd-degree polynomials with real coefficients have at least one real root. The graph of a cubic function always has a single inflection point. It may have two critical points, a local minimum and a local maximum. Otherwise, a cubic ...
A small change of coefficients may induce a dramatic change of the roots, including the change of a real root into a complex root with a rather large imaginary part (see Wilkinson's polynomial). A consequence is that, for classical numeric root-finding algorithms , the problem of approximating the roots given the coefficients can be ill ...
Finding roots −1/2, −1/ √ 2, and 1/ √ 2 of the cubic 4x 3 + 2x 2 − 2x − 1, showing how negative coefficients and extended segments are handled. Each number shown on a colored line is the negative of its slope and hence a real root of the polynomial. To employ the method, a diagram is drawn starting at the origin.
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
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:
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