Sunday, May 29, 2016

HP Prime: Basic CAS Commands for Polynomials and Rational Expressions

HP Prime:  Basic CAS Commands for Polynomials and Rational Expressions

Define the following variables:

poly:  a polynomial of the form a_n*x^n + a_n-1*x^(n-1) + …. + a_1*x + a_0
rat:  a rational function consisting of the polynomials p(x)/q(x)
var: variable

Simplification Modes

Before we start, I want to comment on simplification modes.  

Simplification Modes:

None:  No simplification is executed
Minimum:  Simple simplification of results.  Additional simplification may be desired.
Maximum:  Full simplification of results

Below is a comparison between Minimum and Maximum modes.

To change Simplification mode, press [Shift], [CAS] (CAS Settings), and select a Simplification settings.  Select the Simplification drop down box and select the desired mode.    

Note:  The following examples are executed in Maximum simplification mode.

Part Extraction:  Coefficients, Numerator, Denominator


coef:  [Toolbox], (CAS), 6. Polynomial, 2.  Coefficients

coef(poly, var):  returns all the coefficients of a polynomial in a vector
coef(poly, var, n): return the coefficient of a polynomial in a vector of the specific power x^n

coef(2x^2 + 3x – 1, x) returns [2, 3, -1]
coef(2x^2 + 3x – 1, x, 2) returns 2
coef(2x^2 + 3x – 1, x, 1) returns 3
coef(2x^2 + 3x – 1, x, 0) returns -1

Numerator and Denominator

numer: [Toolbox], (CAS), 1. Algebra, 8. Extract, 1.  Numerator
denom: [Toolbox], (CAS), 1. Algebra, 8. Extract, 2. Denominator


numer(f) returns 2*x^2 – 3
denom(f) returns x^2 + 1
purge(f) \\ this is to erase f

Polynomial Creation

Create a symbolic polynomial with a list of coefficients

poly2symb: [Toolbox], (CAS), 6. Polynomial, 7.  Create, 2. Coefs→Poly


poly2symb(vector of coefficients, var)


poly2symb( [8, -1, 0, 6], t) returns 8*t^3 – t^2 + 6

The inverse operation is the symb2poly(poly), which can be accessed by [Toolbox], (CAS), 6. Polynomial, 7. Create, 1. Poly→Create. 

Create a polynomial from a list of roots

This involves a two-step processes:
1. pcoeff( vector of roots )
2. poly2symb( result from step 1, var )

Access pcoeff: [Toolbox], (CAS), 6. Polynomial, 7. Create, 3.  Roots → Coef

You can combine the two steps by typing:  poly2symb(pcoeff( vector of roots), var).  Simplification may be required


pcoeff([-2, 5, 0, 6]) returns [1, -9, 8, 60, 0]
poly2symb([1, -9, 8, 60, 0],t) returns t^4 – 9*t^3 +8*t^2 + 60*t

poly2symb( pcoeff([2, -5, 0, 6]), t) returns 
t^4 – 9*t^3 + 8*t^2 + 60*t

Degree of a Polynomial

degree: [Toolbox], (CAS), 6, Polynomial, 8. Algebra, 3. Degree



f: = 4*x^3 – 2*x^2 + 8*x – 8
degree(f) returns 3

You can factor a polynomial by x^n where n is the degree of the polynomial by using factor_xn.

factor_xn:  [Toolbox], (CAS), 6. Polynomial, 8. Algebra, 4. Factor By Degree
Caution:  This command works when the Simplification mode is turned to Minimum or Off.    

(after turning Simplification to Minimum)
factor_xn(f) returns x^3 * (4 – 2/x + 8/x^2 – 8/x^3)

Partial Fraction of Rational Functions

partfrac: [Toolbox], (CAS), 1. Algebra, 7. Partial Fraction




partfrac( (x^4 – 3*x^3)/(x^2 -1) ) returns 
x^2 -3*x + 1 - 1/(x-1) - 2/(x+1)

Determining the Number of Zeros (Roots)

The sturmab command determines the number of zeros giving an interval.

sturmab:  [Toolbox], (CAS), 6. Polynomial, 8. Algebra, 6.  Zero Count


sturmab(poly, var, min, max)

The min and max can be complex numbers.


sturmab(x^3 – 4*x^2 + 6*x – 4, x, -5, 5) returns 1  \\ 1 real root
sturmab(x^3 – 4*x^2 + 6*x – 4, x, -5-5*i, 5+5*i) returns 3  \\ 1 real, 2 complex roots

Polynomial Functions

Path: [Toolbox], (CAS), 6, Polynomial, 9. Special, then:
4.  Hermite \\ Syntax:  hermite(n)
7.  Legendre \\ Syntax:  legendre(n)
8.  Chebyshev Tn (1st Kind)  \\ Syntax:  tchebyshev1(n)
9.  Chebyshev Un (2nd Kind)  \\ Syntax:  tchebyshev2(n)

Where n is an integer.  You can specify a variable by adding var as a second argument. (x is the default variable)

hermite(4) returns 16*x^4 – 48*x^2 + 12
legendre(4) returns (35*x^4 – 30*x^3 + 3)/8
tchebyshev1(4) returns 8*x^4 – 8*x^2 + 1
tchebyshev2(4) returns 16*x^4 – 12*x^2 + 1

This covers some of the basic CAS commands for polynomials and rational functions.  Hope you find this helpful,


This blog is property of Edward Shore, 2016.


  1. Really i am impressed from this post....the person who created this post is a genius and knows how to keep the readers connected..

    HP Printer Phone Number

  2. Our riddle ended up being We want to writing adapt the actual amazing essential on Hormone balance. Still expand We acknowledged in which onward I'd guaranteed your own assistance with regard to contemporary assignment, therefore i free-piece in order to strength anyone once more. right triangle calculator

  3. This comment has been removed by a blog administrator.


Numworks Update 17.2 - Highlights

 Numworks Update 17.2 - Highlights Update is available at: The Numworks website will detect your calculator and wi...