**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**

**Coefficients**

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

Syntax:

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

Examples:

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

Syntax:

numer(rat)

denom(rat)

Example:

f≔(2x^2-3)/(x^2+1)

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

Syntax:

poly2symb(vector of coefficients, var)

Example:

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

Example:

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

Syntax:

degree(poly)

Example:

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.

Example:

(after turning Simplification to Minimum)

factor_xn(f) returns x^3 * (4 – 2/x + 8/x^2 – 8/x^3)

purge(f)

**Partial Fraction of Rational Functions**

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

Syntax:

partfrac(rat)

Example:

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

Syntax:

sturmab(poly, var, min, max)

The min and max can be complex numbers.

Examples:

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 (1

^{st}Kind) \\ Syntax: tchebyshev1(n)
9. Chebyshev Un (2

^{nd}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)

Examples:

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,

Eddie

This blog is property of Edward Shore, 2016.

‘

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

ReplyDeleteHP Printer Phone Number

Thank you. It is very useful.

ReplyDeleteThank you!

ReplyDeleteI am impressed too...

ReplyDeleteOur 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

ReplyDelete