Solving cubic equations

Solving Cubic Equations

Definition of Cubic Equations

  • A cubic equation is a type of polynomial equation that appears in the form ax³ + bx² + cx + d = 0, where a, b, c, and d are coefficients and ‘a’ is non-zero.

Understanding the Graph of Cubic Functions

  • The graph of a cubic function is called a cubic curve, or a cubic polynomial. It can have one local minimum and one local maximum, or it can be monotonic, with none of these. The curve is continuous and smooth, without sharp turns or corners.
  • It also has an end behaviour similar to a quadratic function, meaning that as x approaches infinity, the y values of the graph either will both approach infinity (positive cubic) or one will approach positive infinity while the other approaches negative infinity (negative cubic).
  • The graph of a cubic function will cross the x-axis at the solutions of the cubic equation.

Techniques for Solving Cubic Equations

  • If the cubic equation is straightforward, the trial and error method can be used to find one root, and then simplify the equation into a quadratic.
  • Factorising can also be used if the cubic equation can be rearranged into a trinomial form. Then it can be factorised and solved as a quadratic equation.
  • The technique for solving cubic equations is generally to first find one root, then to factor the cubic down to a quadratic using polynomial division, then solve the remaining quadratic equation.

The Nature of Roots

  • A cubic equation always has at least one real root, but it can also have two or three real roots.
  • The solutions of a cubic equation might be real or complex. The nature of roots can be determined without solving the equation by using the concept of the discriminant.
  • If the discriminant is greater than 0, all roots are real and different. If it is equal to 0, one root is real and different, while the other two are real and equal. If the discriminant is less than 0, all roots are real and different.

Applications of Cubic Equations

  • Cubic equations appear in various applications in physics, engineering, and economics to model certain conditions or behaviors.
  • For example, they can be used to describe the flow rate of certain fluids, understand population growth, or analyze certain economic trends.