Saturday, November 18, 2023

HP 32S and HP 15C: Synthetic Division

HP 32S and HP 15C:   Synthetic Division



These programs perform synthetic division, where a polynomial of degree n is divided by the binomial term x - x_0:


[a_n × x^n + a_(n-1) × x^(n-1) + a_(n-2) × x^(n-2) + ... a_1 × x + a_0] ÷ (x - x_0)


With the result of the division is:


[q_(n-1) × x^(n-1) + q_(n-2) × x^(n-2) + ... + q_1 × x + q_0] + R ÷ (x - x_0)


where: 


q_(n-1) = a_n

R is the remainder.



HP 32S/32SII:  Synthetic Division


The degree of the polynomial can go up to 24, with the variables set up as:


p(x) = [ A + B × x + C × x^2 + D × x^3 + ... + Y × x^24 ] ÷ (x - Z)


where

A is the constant

B is the coefficient of x

C is the coefficient of x^2 

and so on...

Z = x_0


Flag 1 is used as an indicator that there are coefficients q_k.  



Code:


S01  LBL S

S02  CF 1

S03  1

S04  +

S05  STO i

S06  SF 1

S07  RCL (i)

S08  STOP

S09  DSE i


Z01  LBL Z

Z02  RCL× Z

Z03  RCL+ (i)

Z04  STOP

Z05  DSE i

Z06  GTO Z

Z07  CF 1

Z08  RTN


Total Memory:  S:  13.5 bytes, Z:  12.0 bytes, Total:  25.5 bytes


To run:  store the coefficients and x0.  Enter the degree of the polynomial and press XEQ S.  



HP 15C:  Synthetic Division


The degree of the polynomial can go as high as you want, as long as you have registers.   The variables set up as:


p(x) = [ R0 + R1 × x + R2 × x^2 + R3 × x^3 + ... ] ÷ (x - R0)


where

R0 is the constant

R1 is the coefficient of x

R2 is the coefficient of x^2 

and so on...

R0 = x_0


Flag 8 is used as an indicator (C on the screen) that there are coefficients q_k.  In effect, this turns on complex mode, which would create a stack of imaginary numbers.  The complex mode for the HP 15C requires five registers to run.  


Code:


001  LBL A:   42,21,11

002  CF 8:  43, 5, 8

003  1:  1

004  +:  40

005  STO I:  44, 25

006  SF 8:  43, 4, 8

007  RCL (i):  45, 24

008  R/S:   31

009  DSE I:  42, 5, 25


010  LBL 0:  42, 21, 0

011  RCL× 0:  45, 20, 0

012  RCL+ (i):  45,40,24

013  R/S:  31

014  DSE I:  42, 5,25

015  GTO 0:  22, 0

016  CF 8:  43, 5, 8

017  RTN:  43,32


To run:  store the coefficients and x0.  Enter the degree of the polynomial and press GSB A.  



Examples



Example 1:  [ 4x^3 + 3x^2 + 2x - 5 ] ÷ [ x -  1 ]


HP 32S:

A:  -5,  B:  2,  C:  3,  D:  4,  Z:  1


HP 15C: 

R1:  -5,  R2:  2,  R3:  3,  R4:  4,  R0:  1


Degree: 3


3 XEQ S/GSB A:


Results:  

4   (x^2)  R/S

7   (x)   R/S

9   (constant)  R/S

4  (remainder)


4x^2 + 7x + 9 + 4 ÷ (x-1)



Example 2:  [ x^4 - 3x^2 + 6x + 3 ] ÷ [ x + 3 ]


HP 32S:

A:  3,  B:  6,  C:  -3, D:  0,  E:  1,  Z:  -3


HP 15C:

R1:  3,  R2:  6,  R3:  -3, R4:  0,  R5:  1,  R0:  -3


Degree:  4


4 XEQ S/GSB A


Results:

3   (x^3)  R/S

6   (x^2)  R/S

-3   (x)   R/S

0   (constant)  R/S

1  (remainder)


x^3 - 3x^2 + 6x - 12 + 39 ÷ (x + 3)



Example 3:  [ x^3 - 42x^2 + 395x + 966 ] ÷ [ x - 21 ]


HP 32S:

A:  966,  B:  395,  C:  -42,  D:  1,  Z:  21


HP 15C:

R1:  966,  R2:  395,  R3:  -42,  R4:  1,  R0:  21


Degree:  3


3 XEQ S/GSB A:


Results:  

1   (x^2)  R/S

-21   (x)   R/S

-46   (constant)  R/S

0 (remainder)


x^2 - 21x - 46



Source


Hewlett Packard.  "Synthetic Division"   HP-65 Math Pac 1.   September 1974.   pp. 34-35, 99




Next blog post:  November 25, 2023


Eddie


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