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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.

ObjectiveEdit

  • Present cubic function and cubic equation.
  • Introduce the concept of roots of equal absolute value.
  • Show how to predict and calculate equal roots, techniques that will be useful when applied to higher order functions.
  • Simplify the depressed cubic.
  • Simplify Vieta's substitution.
  • Review complex numbers as they apply to a complex cube root.
  • Show that the cubic equation is effectively solved when at least one real root is known.
  • Use Newton's Method to calculate one real root.
  • Show that the cubic equation can be solved with high-school math.

LessonEdit

IntroductionEdit

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:

  • both coefficients   must be non-zero,
  • coefficient   must be positive (simply for our convenience),
  • all coefficients must be real numbers, accepting that the function may contain complex roots.

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 functionsEdit

The function may be expressed as:

  where   are roots of the function, in which case

  where:  

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 quadraticEdit

When   is a root of the function, the function may be expressed as:

  where

 

When one real root   is known, the other two roots may be calculated as roots of the quadratic function  .

The simplest cubic functionEdit

 
Figure 1.

The simplest cubic function has coefficients  

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 valueEdit

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  :

 

From  

If   is a solution and function   becomes:

 

 

  and two roots of   are  .

An exampleEdit

 
Figure 2.

The roots of equal absolute value are  

See Figure 2.


 

 

The function has roots of equal absolute value.

 

The roots of equal absolute value are  .

Equal RootsEdit

Combine   and   from above to eliminate   and produce a function in  :


         


From   above:    


If  , then   is a solution ,   and   become:

 

 

If   because  , there is a stationary point where  .


Note that   is the discriminant of the cubic formula. If   because the discriminant is  , function   contains at least 2 roots equal to   when both functions   are  .

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.


  and   can be combined to produce:

 

 


  and   can be combined to produce:

 

 


If the original function   contains 3 unique roots, then   are numerically different.


If the original function   contains exactly 2 equal roots, then   are numerically identical, and the 2 roots have the value   in  .


If the original function   contains 3 equal roots, then   are both null,   are numerically identical and  .


From equations (4g) and (5g):

     

ExamplesEdit

No equal rootsEdit

 
Figure 3a.

Cubic function with 3 unique, real roots at  .

Consider function  

from  

from  

  are numerically different.

Exactly 2 equal rootsEdit

 
Figure 3b.

Cubic function with 2 equal, real roots at  .

Consider function  

from  

from  

There are 2 equal roots at  

3 equal rootsEdit

 
Figure 3c.

Cubic function with 3 equal, real roots at  .

Consider function  

from  

from  

from  

  are numerically identical, the discriminant of each is   and  

Depressed cubicEdit

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:

  where

  and

 

Be prepared for the possibility that one or both of   may be zero.

When A = 0Edit

 
Figure 4a.

Cubic function with slope 0 at point of inflection  .

This condition occurs when the cubic function in   has exactly one stationary point or when slope at point of inflection is zero.

 

 

 

 

The other roots may be derived from the associated quadratic:

  

       

When B = 0Edit

 
Figure 4b.

Cubic function with point of inflection   on   axis.

This condition occurs when the cubic function in   is of format   or when point of inflection is on the   axis.

 

 

 

 

 

When A = B = 0Edit

 
Figure 4c (same as 3c above).

Cubic function with:
* point of inflection   on   axis,
* slope   at point of inflection.

This condition occurs when:

  • slope at point of inflection is  , and
  • point of inflection is on   axis.


Consider function  

     

 

Vieta's substitutionEdit

Let the depressed cubic be written as:   where   and  

Let  

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.

Discriminant  Edit

 
Figure 5a.

Cubic function with 2 equal, real roots at  .

If discriminant  , the function contains at least 2 equal, real roots.

Consider function   

 

 

 

 

Associated quadratic    

The 2 equal roots are:  .


Discriminant   positiveEdit

 
Figure 5b.

Cubic function with discriminant   positive
and 1 real root at  .

If discriminant   is positive, the function contains exactly 1 real root.

Consider function  

discriminant  

 

 

 

 

or:

 

 

 


 

The associated quadratic is:  

and the two complex roots are:      

Discriminant   negativeEdit

If discriminant   is negative, the function contains 3 real roots and   becomes the complex number  .


Let   be the modulus of  .

Let   be the real part of  .

Let   be the imaginary part of  .

Then  

 

 

 

Let   be the phase of  .

Then   and  .

 . Therefore:

 

 

 

 

An exampleEdit

 
Figure 5c.

Cubic function with 3 unique, real roots at  .

  in which  

 

 

 

 

 

  radians.

  radians.

 

 

 

 

 

Review of complex mathEdit

 
Figure 6a: Components of complex number Z.

Origin at point  .
  parallel to   axis.
  parallel to   axis.
  = modulus of   
Angle   is the phase of    


 
Figure 6b: Complex numbers   and  .

Origin at point  .
  (off image to left.)
 
 
 

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 rectangular format.

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 numbersEdit

       

To multiply complex numbers, multiply the moduli and add the phases.

Complex number cubedEdit

       

Cube root of complex number WEdit

Let   and  

If   then:

  and

 

Complex number  Edit

Let   where  

 

If  

 

In the case of 3 real roots,  

cos Edit

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  Edit

 
Figure 7a.

Graph of  

The well known identity for   is:

 

The derivation of this identity may help understanding and interpreting the curve of  

Let  

  and  

Therefore the point   is on the curve and  

A 3A cos A cos 3A
0 0 1 1
180 180*3 -l -1
60 180 0.5 -1

Three simultaneous equations may be created from the above table:

 

 

Therefore  

 

 

  and  

When   is known,  

Newton's MethodEdit

 
Figure 7b.

Newton's Method used to calculate   when  

Newton's method is a simple and fast root finding method that can be applied effectively to the calculation of   when   is known because:

  • the function is continuous in the area under search.
  • the derivative of the function is continuous in the area under search.
  • the method avoids proximity to stationary points.
  • a suitable starting point is easily chosen.

See Figure 7b.

Perl code used to calculate   when   is:

$cos3A = 0.1;

$x = 1; # starting point.
$y = 4*$x*$x*$x - 3*$x - $cos3A;

while(abs($y) > 0.00000000000001){
    $s = 12*$x*$x - 3; # slope of curve at x.
    $delta_x = $y/$s;
    $x -= $delta_x;
    $y = 4*$x*$x*$x - 3*$x - $cos3A;

    print "                                                                                   
x=$x                                                                                              
y=$y                                                                                              
";
}

print "                                                                                           
cos(A) = $x                                                                                       
";
x=0.9
y=0.116

x=0.882738095238095
y=0.00319753789412588

x=0.882234602936352
y=2.68482638085543e-06

x=0.882234179465815
y=1.89812054962601e-12

x=0.882234179465516
y=-3.60822483003176e-16

cos(A) = 0.882234179465516

When   is positiveEdit

 
Figure 7c.

Newton's Method used to calculate   when  

When   output of the above code is:

x=0.933333333333333
y=0.0521481481481486

x=0.926336712383224
y=0.000546900278781126

x=0.926261765753783
y=6.24370847246425e-08

x=0.926261757195518
y=7.7715611723761e-16

cos(A) = 0.926261757195518

If all 3 values of   are required, the other 2 values can be calculated as roots of the associated quadratic function with coefficients  

x1 = -0.136742508909433
x2 = -0.789519248286085

When   is negativeEdit

 
Figure 7d.

Newton's Method used to calculate   when  

When   is negative, the starting value of  

When   output of the above code is:

x=-0.911111111111111
y=-0.0920054869684496

x=-0.89789474513884
y=-0.00190051666894692

x=-0.897610005610658
y=-8.73486682706481e-07

x=-0.89760987462259
y=-1.8446355554147e-13

x=-0.897609874622562
y=-1.66533453693773e-16

cos(A) = -0.897609874622562

An exampleEdit

 
Figure 7e.

Cubic function with 3 unique, real roots at  .

  in which  

 

 

 

 

 

Use the code beside Figure 7b above with initial conditions:

$cosWphi = -0.338086344651354;

$cos3A = $cosWphi;

$x = -1; # starting point.

 

 

 

 

 

Point of InflectionEdit

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:

 

Depressed cubicEdit

Recall from "Depressed cubic" above:

 

 

Therefore:

 

 

If 1 of   is zero, the cubic equation may be solved as under "Depressed cubic" above.

Newton's MethodEdit

If both   of the depressed function are non-zero, Newton's method may be applied to the original cubic function, and the Point of Inflection offers a convenient starting point.

slope at PoI positiveEdit

 
Figure 8a.

 
Cubic function with positive slope at Point of Inflection  

 

 

 

 

 

slope at PoI negativeEdit

PoI above X axisEdit

 
Figure 8b.

 
Cubic function with negative slope at Point of Inflection  
and PoI above   axis.

 

 

 

 

 

When   the other 2 intercepts may be calculated as roots of the associated quadratic with coefficients:

     

 

($a,$b,$c,$d) = (0.1,0.9,0.2,-2.8);
($x,$y) = ($x1a,$ypoi);

while(abs($y) > 1e-14){
    $s = 3*$a*$x*$x + 2*$b*$x + $c;
    $delta_x = $y/$s;
    $x -= $delta_x ;
    $y = $a*$x*$x*$x + $b*$x*$x + $c*$x + $d;
    print "
x=$x,y=$y
";
}

print "
x=$x
";
x=-8.4,y=-0.246400000000004

x=-8.36056338028169,y=-0.00251336673084257

x=-8.36015274586958,y=-2.7116352896428e-07

x=-8.36015270155726,y=-8.88178419700125e-16

x=-8.36015270155726

PoI below X axisEdit

 
Figure 8c.

 
Cubic function with negative slope at Point of Inflection  
and PoI below   axis.

 

 

 

 

 

When   the other 2 intercepts may be calculated as roots of the associated quadratic with coefficients: