**DM 41L and HP 41C: Generating a Polynomial Given Its Roots**

**Introduction**

Generate the coefficients of a polynomial (up to the order 4) with the roots a_0, a_1, a_2, and a_3. The resulting polynomial is:

p(x) = (x – a_0) * (x – a_1) * (x – a_2) * (x – a_3) * (x – a_4)

p(x) = r_4 * x^4 + r_5 * x^3 + r_6 * x^2 + r_7 * x + r_8

The default is a polynomial where the lead coefficient is positive. If you want a polynomial where the lead coefficient is negative, multiply every coefficient by -1.

**Instructions**

Store the four roots in registers R00, R01, R02, and R03 respectively. Run POLY4. Coefficients are shown briefly as they are calculated. They are can be recalled by the registers in decreasing order of x: R04, R05, R06, R07, and R08.

**DM 41L and HP 41C Program: POLY4**

01 LBL^T POLY4

02 1

03 STO 04

04 PSE

05 RCL 00

06 CHS

07 RCL 01

08 –

09 RCL 02

10 –

11 RCL 03

12 –

13 STO 05

14 PSE

15 RCL 01

16 RCL 02

17 +

18 RCL 03

19 +

20 RCL 00

21 *

22 RCL 02

23 RCL 03

24 +

25 RCL 01

26 *

27 +

28 RCL 02

29 RCL 03

30 *

31 +

32 STO 06

33 PSE

34 RCL 01

35 RCL 02

36 *

37 RCL 01

38 RCL 03

39 *

40 +

41 RCL 02

42 RCL 03

43 *

44 +

45 RCL 00

46 *

47 CHS

48 RCL 01

49 RCL 02

50 *

51 RCL 03

52 *

53 –

54 STO 07

55 PSE

56 RCL 00

57 RCL 01

58 *

59 RCL 02

60 *

61 RCL 03

62 *

63 STO 08

64 RTN

Example

Roots x = -3, x = 3, x= 4, and x= 6

Coefficients:

R04 = 1

R05 = -10

R06 = 15

R07 = 90

R08 = -216

Polynomial: p(x) = x^4 – 10 * x^3 + 15 * x^2 + 90 * x – 216

Eddie

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