DM41X Polynomial Solver

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====== DM41X : Polynomial Solver ======

** Advantage Pac Polynomials **

Assuming ''Advantage Pac'' is //plugged in//...


Finds all roots, including complex ones.

==== Save to registers ====

To save the discovered roots in registers

  * Enter the polynomial coefficients as normal
  * prior to pressing ''RT'' to do the solve **SET FLAG 06**
    * <key>Shift</key><key>SF</key>''06''
  * This will save the discovered roots in ''R24'' upwards
    * saves in pairs as ''REAL'' and ''IMAGINARY''
      * ''R24'' = ''REAL root 1''
      * ''R25'' = ''IMAG root 1''
      * ''R26'' = ''REAL root 2''
      * ''R27'' = ''IMAG root 2''
      * etc
      * etc



===== Roots of a quadratic =====

To find the roots of a simple quadratic (with 2 real roots)

'' x<sup>2</sup> - 3x + 2 = 0 ''

  * <key>XEQ</key> <key>ALPHA</key>PLY<key>ALPHA</key>
  * Asks for ''DEGREE=?''
    * this is the highest power of x in the polynomial
    * enter ''2'' <key>R/S</key>
  * Asks for ''a2=?''
    * the coefficient of ''x<sup>2</sup>''
    * enter ''1'' <key>R/S</key>
  * Asks for ''a1=?''
    * the coefficient of ''x''
    * enter ''3'' <key>CHS</key>  <key>R/S</key>
  * Asks for ''a0=?''
    * the constant term
    * enter ''2'' <key>R/S</key>
  * Display shows ''FX  RT    NEW''
    * ''FX'' = Function value at your choice of ''x''
    * ''RT'' = Root(s) of the entered polynomial
    * ''NEW'' = Enter a new polynomial
    * Choose ''RT'' <key>'B'</key>
  * Display shows ''ROOT = 2.0000''
    * to see next root press <key>R/S</key>
  * Display shows ''ROOT = 1.0000''

The roots of our quadratic are ''2'' and ''1''

===== Higher order with complex roots =====

Either press <key>TAN</key> (''J'') to get the menu, and then ''NEW'' (<key>LN</key>)  or XEQ PLY again

'' 4x<sup>4</sup> - 8x<sup>3</sup> - 13x<sup>2</sup> - 10x + 22 = 0 ''

This should have 4 roots, and most likely some will be complex.

  * ''DEGREE=?''
    * ''4'' <key>R/S</key>
  * ''a4=?''
    * ''4'' <key>R/S</key>
  * ''a3=?''
    * ''8''<key>CHS</key><key>R/S</key>
  * ''a2=?''
    * ''13'' <key>CHS</key><key>R/S</key>
  * ''a1=?''
    * ''10'' <key>CHS</key><key>R/S</key>
  * ''a0=?''
    * ''22''<key>R/S</key>
  * ''FX RT     NEW''
    * ''RT'' <key>'B'</key>

Roots are displayed:

  * ''U = -1.0000'' <key>'R/S'</key> 
  * ''V = 1.0000'' <key>'R/S'</key> 
  * ''U = -1.0000'' <key>R/S</key>
  * ''V = -1.0000'' <key>R/S</key>
  * ''ROOT = 3.1180'' <key>R/S</key>
  * ''ROOT = 0.8820'' <key>R/S</key>
  * ''FX RT    NEW''

The 4 roots are 
  * ''-1 + j 1 ''
  * '' -1 - j 1 ''
  * '' 3.1180 ''
  * '' 0.8820 ''


===== Cube Roots of minus 8 =====

If x is the cube root of minus 8 then '' x<sup>3</sup> = -8 '' and as a polynomial '' x<sup>3</sup> + 8 = 0 ''

Simply use ''0'' as the coefficient of X<sup>2</sup> and x

  * <key>XEQ</key> <key>ALPHA</key>PLY<key>ALPHA</key>
  * ''Degree=?''   ''3'' <key>R/S</key>
  * ''a3=?'' ''1'' <key>R/S</key>
  * ''a2=?'' ''0'' <key>R/S</key>
  * ''a1=?'' ''0'' <key>R/S</key>
  * ''a0=?'' ''8'' <key>R/S</key>

  * ''RT'' <key>'B'</key>
    * ''ROOT = -2.0000'' <key>R/S</key>
    * ''U = 1.0000''  <key>R/S</key>  ''V =  1.7321''
    * ''U = 1.0000''  <key>R/S</key>  ''V = -1.7321''

  * Roots are 
    * '' 2.0000 ''
    * '' 1 + j 1.7321 ''
    * '' 1 - j 1.7321 ''





--- //John Pumford-Green 26/01/26 09:55 GMT//


===== Further Information =====


{{tag>calculator dm41x}}

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