It is not the purpose of this page to repeat good information available elsewhere. However, it seems to the author that other descriptions of the cubic function are more complicated than they need to be. This page attempts to demystify elementary but essential information concerning the cubic function.
- 1 Objective
- 2 Lesson
- 3 The simplest cubic function
- 4 Roots of equal absolute value
- 5 Equal Roots
- 6 Depressed cubic
- 7 Vieta's substitution
- 8 Review of complex math
- 9 cos
- 10 Point of Inflection
The cubic function is the sum of powers of from through :
usually written as:
If the function becomes
Within this page we'll say that:
The cubic equation is the cubic function equated to zero:
Roots of the function are values of that satisfy the cubic equation.
Function as product of 3 linear functions
The function may be expressed as:
where are roots of the function, in which case
Solving the cubic equation means that, given , at least one of must be calculated.
Given , I found that can be calculated as:
This approach was not helpful.
Function as product of linear function and quadratic
When is a root of the function, the function may be expressed as:
When one real root is known, the other two roots may be calculated as roots of the quadratic function .
The simplest cubic function
The simplest cubic function has coefficients , for example:
To solve the equation:
The function also contains two complex roots that may be found as solutions of the associated quadratic:
Roots of equal absolute value
The cubic function
Let one value of be and another be .
Substitute these values into the original function in and expand.
Reduce and and substitute for :
Combine and to eliminate and produce a function in :
If is a solution and function becomes:
and two roots of are .
See Figure 2.
The function has roots of equal absolute value.
The roots of equal absolute value are .
Combine and from above to eliminate and produce a function in :
If because , there is a stationary point where .
Note that are functions of the curve and the slope of the curve. In other words, equal roots occur where the curve and the slope of the curve are both zero.
No equal roots
Exactly 2 equal roots
3 equal roots
The depressed cubic may be used to solve the cubic equation.
In the cubic function: let , substitute for and expand:
In the depressed function the coefficient of is and the coefficient of is .
When the function is equated to , the depressed equation is:
Be prepared for the possibility that one or both of may be zero.
When A = 0
When B = 0
When A = B = 0
Let the depressed cubic be written as: where and
Substitute for in the depressed function:
where and .
From the quadratic formula:
The discriminant . Substitute for and expand:
This discriminant =
The factor is a factor of above.
Review of complex math
A complex number contains a real part and an imaginary part, eg:
In theoretical math the value is usually written as . In the field of electrical engineering and computer language Python it is usually written as .
The value is a complex number expressed in polar format where is the modulus of or and is the phase of or
Multiplication of complex numbers
To multiply complex numbers, multiply the moduli and add the phases.
Complex number cubed
Cube root of complex number W
Complex number 
In the case of 3 real roots,
The method above for calculating depends upon calculating the value of angle
However, may be calculated from because
Generally, when is known, there are 3 possible values of the third angle because
This suggests that there is a cubic relationship between and
Expansion of 
Point of Inflection
The Point of Inflection is the point at which the slope of the curve is minimum.
After taking the first and second derivatives value at point of inflection is:
The slope at point of inflection is:
Value at point of inflection is:
Recall from "Depressed cubic" above:
If 1 of is zero, the cubic equation may be solved as under "Depressed cubic" above.