Solving cubic or quartic equations with real coefficients

A cubic equation is a type of polynomial equation where the highest exponent power is three, leading to a maximum of three roots.
Similarly, a quartic equation is a polynomial equation with the highest exponent of four, allowing for four possible solutions.
Solving cubic or quartic equations typically involves the Factor Theorem and Rational Root Theorem, testing potential root candidates.
An alternative technique is the method of completing the cube and completing the quartic.

Contrary to popular belief, cubic and quartic equations with real coefficients aren’t limited to real solutions; they can also yield complex solutions.
This is because of the Fundamental Theorem of Algebra, which states every non-constant polynomial equation, regardless of coefficients being real or complex, has at least one complex root.

Solving cubic equations often begins by applying synthetic division or the rational root test to find a real root, using factor theorem to find the linear factor.
Once a linear factor is obtained, the cubic equation can be factored into a linear term and a quadratic term, the latter of which can be solved using the quadratic formula.
Remember the quadratic formula might give rise to complex roots in the form a ± bi.

Solving quartic equations can also start with the rational root test to find a root, factoring the depressed quartic, and then solving a cubic equation.
If a quartic equation can be factored into two quadratic equations, it can be solved quicker with the quadratic formula.
In some cases, the method of solving cubic equations can be applied to reduce a quartic to a quadratic.
Like cubic equations, the solutions of quartic equations also can yield complex roots.

Solving cubic and quartic equations has practical use in diverse fields like engineering, physics and mathematics.
In physics, cubic equations often appear within the context of motion under constant acceleration, while quartic equations can be used to describe some types of movement in two dimensions.
Understanding and proficiency in solving cubic and quartic equations with real coefficients fosters a deep comprehension of algebraic structures and symmetry, useful in advanced mathematics.