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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