Questions tagged [cubics]

A cubic equation has the form $$ax^3 + bx^2 + cx + d = 0 $$ where $~a,~b,~c,~d~$ are complex numbers and $~a \ne 0~.$

By the Fundamental Theorem of Algebra, cubic equation always has $~3~$ roots, some of which might be equal. All cubic equations have either one real root, or three real roots.

All of the roots of the cubic equation can be found algebraically. The roots can also be found trigonometrically. Alternatively, numerical approximations of the roots can be found using root-finding algorithms such as Newton's method.

Applications:

Cubic equations arise in various other contexts.

• Marden's theorem states that the foci of the Steiner inellipse of any triangle can be found by using the cubic function whose roots are the coordinates in the complex plane of the triangle's three vertices. The roots of the first derivative of this cubic are the complex coordinates of those foci.

• The area of a regular heptagon can be expressed in terms of the roots of a cubic. Further, the ratios of the long diagonal to the side, the side to the short diagonal, and the negative of the short diagonal to the long diagonal all satisfy a particular cubic equation. In addition, the ratio of the inradius to the circumradius of a heptagonal triangle is one of the solutions of a cubic equation. The values of trigonometric functions of angles related to $~{\displaystyle 2\pi /7}~$ satisfy cubic equations.

• Given the cosine (or other trigonometric function) of an arbitrary angle, the cosine of one-third of that angle is one of the roots of a cubic.

• The solution of the general quartic equation relies on the solution of its resolvent cubic.

• 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 linear difference equation or differential equation is a cubic equation.

• In analytical chemistry, the Charlot equation, which can be used to find the pH of buffer solutions, can be solved using a cubic equation.

• In chemical engineering and thermodynamics, cubic equations of state are used to model the PVT (pressure, volume, temperature) behavior of substances.

• Kinematic equations involving changing rates of acceleration are cubic.

• The speed of seismic Rayleigh waves is a solution of the Rayleigh wave cubic equation.

References:

https://en.wikipedia.org/wiki/Cubic_function

http://mathworld.wolfram.com/CubicFormula.html

http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-cubicequations-2009-1.pdf