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Polynomial Toolkit for TI-Nspire

## Polynomial Toolkit for TI-Nspire

It is possible to perform some symbolic computations involving polynomials on the numeric (non-CAS) version of TI-Nspire, effectively supporting functionality normally expected of a computer algebra system (CAS). The polynomial toolkit is a collection of utility programs and functions intended to assist in the teaching and learning of high school algebra by making it easier to create and use documents that involve algebraic operations. They simulate some of the symbolic functionality usually associated with CAS, but work equally well in both CAS and numeric TI-Nspire environments. If the document "poly.tns" (attached) is placed within the library (MyLib) folder, then these are accessible in new documents.

The toolkit contains three types of elements:

Functions (which can be used anywhere within TI-Nspire, from Notes to Graphs);

Programs which make use of dialog boxes for input and display, and so need to be run from the Calculator; and

Quizzes (which, like the programs, will need to be run from the Calculator).

Functions

Functionstake an algebraic input and operate upon it to return a result (which is always a string). For example,simp("(2x-1)(3x+2)") > "6x^2+1x^1+-2"

pfactor(2*x^2-6*x+4) > "2(x-2)(x-1)"

xpress(der((2*x-3)*(x^2+1))) > "6x

^{2}-6x+2"Functions will always return some object, and they may be used in any TI-Nspire application (Notes, L&S, Calculator - even in Graphs & Geometry). They may also be nested. In the previous example, the function

deerreturned the derivative of the polynomial, thenxpresstook that string and presented it in correct format.While you can enter input as a string, if you run the programs

pvar(var)orpolyvar()to define a variable, then students may enter the input without quotes exactly as they would normally do. For example, pvar("m") will set the variable to "m" and expressions involving this variable may be entered without quotes. Run these in Calculator.These polynomial functions will work up to degree 6.

F1. xpress(poly)- correctly displays a defined polynomial function. (also pdisp(poly)). eg xpress("x^2-4") > "x

^{2}-4"F2. simp(poly)- gives simplified form of polynomial (also pexpand(poly)).

F3. pdegree(poly)- returns the degree (as a number) of polynomial (up to degree 6).

F4. pfactor(poly)- gives the factored form of polynomial.

F5. prand(degree)- generates random polynomials up to the degree specified. Use with xpress()for correct format.

F6. proots(poly)- returns all roots of polynomials up to degree 6. (also psolve(poly)).

F7. der(poly)- computes symbolic derivatives of polynomials.

F8. integ(poly)- computes symbolic integrals of polynomials.

F9. commonfactor(poly)- returns the common factor of a polynomial.

F10. conic(poly)- solves conics in x and y [use pvar("x,y")to set up variables] OR with list of five points.

F11. cplot("poly")- used to plot conics: eg expr(cplot("2x^2-4x-3y^2-12y-12"))[MUST enter as a string in G&G) OR with list of five points.

F12. cts(poly) - returns the completed square form for a quadratic expression or equation.

F13. pcheck(poly1, poly2)- checks equality of poly1 and poly2 (expressions or equations).

F14. pcoeffs(poly) returns the list of coefficients for a polynomial. eg pcoeffs(2*x-3) -> {-3,2,0,0,0,0,0}

F15. pdivide(poly1,poly2) returns the polynomial division result for the two polynomials. eg pdivide(x

^{3}-x-6,x-3)F16. ptermlist(poly1) returns the list of terms for a polynomial. eg ptermlist(x

^{3}-x-6) -> {x^{3},-x,-6}

Utility Functions:F17. pterm(list)- takes the list of coefficients output by pcoeffs and returns the polynomial in x. eg pterm(pcoeffs(2*x-3)) > "2.x^1+-3."

F18. pterms(list,var)- like pterm but specify the required variable as a string: eg pterms(pcoeffs(2*x-3),"w") > "2.w^1+-3."

F19. projectile(x0, y0, initial_velocity, initial_angle, gravity) - gives the step-by-step derivation of the trajectory equation for projectile motion in x and y.

F20. factors(num) - returns the list of prime factors for a number.

F21. radical(num) - expressed the simplified radical form for the square root of num.

ProgramsThese programs run best from the Calculator since they use dialogs for input and display. They prompt for a polynomial, display the result and store both as strings.

P1. polyxpress()- correctly displays a defined polynomial function. (also polydisp()).

P2. polysimp()- gives simplified form of polynomial (also polyexpand()).

P3. polydegree()- returns the degree (as a number) of polynomial (up to degree 6).

P4. polyfactor()- gives the factored form of polynomial.

P5. polyrand()- generates random polynomials up to the degree specified.

P6. polysolve()- solves polynomials up to degree 6.

P7. polyder()- computes symbolic derivatives of polynomials.

P8. polyint()- computes symbolic integrals of polynomials.

P9. polycf()- returns the common factor of a polynomial.

P10. conics()- solves conics in x and y OR with list of five points.

P11. completesquare() - returns the completed square form for a quadratic expression or equation.

P12. polycheck()- checks equality of poly1 and poly2 (expressions or equations).

P13. polycoeff() returns the list of coefficients for a polynomial. eg pcoeffs(2*x-3)

F14. polydivide() returns the polynomial division result for the two polynomials.

F15. polytermlist(poly1) returns the list of terms for a polynomial. eg polytermlist(x

^{3}-x-6) -> {x^{3},-x,-6}

Utility Programs:P16. algtiles(mode) - Creates xlist and ylist for algebra tile representation for function in x using scatterplot. Mode 0 is in standard form; mode 1 in factored form. Run in Notes after graphing function in G&G.

P17. eqn_builder() - Use with algtilesleft() and algtilesright() in Notes to set up side-by-side windows. algtilesleft() generates lists xlistl, ylistl and algtilesright() generates xlistr, ylistr. eqn_builder() takes the user step-by-step through the solution process for linear functions.

QuizzesRun these programs from the Calculator and follow the instructions to work through the various algebraic skills step-by-step, or to test ability at different levels of difficulty.

Step-by-Step Quizzes:

Q1. algtilesquiz() - Use with algebra tiles or alone. Tests skill in substituting into expressions. Difficulty levels 1-3.

Q2. ctsquiz()- begin with ctsquiz() then follow the step-by-step process by entering the values requested.

Q3. derivativequiz()- tests ability to compute symbolic derivatives. Difficulty levels 1-3.

Q4. expandquiz()- tests ability to expand polynomials. Difficulty levels 1-3.

Q5. eqnquiz() - enter 1 for a random linear equation in x, or enter your own. You will be prompted to enter each step of the solution.

Q6. factor_pair(num) - displays the prime factors for a number, then asks the user to identify factor pairs until the simplified radical form is displayed.

Q7. factorquiz()- tests ability to factor polynomials. Difficulty levels 1-3.

Q8. simult_subst() - generates random systems of equations in x and y, then quizzes the user at each step of the solution by substitution.

Q9. simplify() - enter 0 to end or 1-3 for level of difficulty. You will be prompted for the simplified form of a random expression in x.

NOTE that these functions and programs return their result strings in unsimplified form (as "2x^3+-3x^2" rather than "2x

By first using the^{3}-3x^{2}") since this form can be used within the spreadsheet or even within Graphs & Geometry, using the "expr" command. For example, it should be possible to graph f1(x) =expr(der("2x^3-3x^2")).xpressfunction, the output can easily be converted to the more pleasing correct form.

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Polynomial Toolkit for TI-Nspire