Polynomials extension for Numbas

This extension provides a new data type and some functions to deal with polynomials

JME data type

This extension adds a new JME data type Numbas.jme.types.polynomial, representing a polynomial in a given variable.

JME functions

polynomial(expression in one variable)

Create a polynomial, automatically detecting the variable name from the expression. This is quite strict about what it accepts - only one variable name, and coefficients and degrees have to be literal numbers, not calculations or references to other variables.

You can either write a literal expression, or pass a string. Note that if you use a literal expression, variables defined in the scope are substituted in. It's safer to use a string.

Examples

• polynomial(x^2-2x+3)
• polynomial("5*x^4 + 2*x")

polynomial(variable_name,coefficients)

Create a polynomial in the given variable, with the given coefficients (coefficients[i] is the coefficient of variable_name^i). Example: polynomial(x,[-1,0,1]) represents the polynomial x^2-1.

mod_polynomial(expression,m) or mod_polynomial(variable_name,coefficients,m)

As above, but all operations on this polynomial will be calculated modulo m.

p1+p2

Add two polynomials

p1+n or n+p1

Add a constant to a polynomial - more convenient than p+polynomial(n).

p1-p2

Subtract p2 from p1

p1-n or n-p1

Subtract a constant from a polynomial (or vice versa) - more convenient than p-polynomial(n).

p1*p2

Multiply two polynomials

p1*n or n*p1

Multiply a polynomial by a constant - more convenient than p*polynomial(n).

p^n

Take polynomial p to the nth (integer, non-negative) power.

quotient(p1,p2)

Divide p1 by p2, and throw away the remainder (polynomial quotient of p1 and p2)

remainder(p1,p2)

Remainder when dividing p1 by p2.

mod(p,n)

Take each coefficient of p mod n.

degree(p)

Degree of p - highest power of the variable with a non-zero coefficient.

p1=p2

Are p1 and p2 equal? True if all the coefficients match.

p[d]

Coefficient of x^d in p.

eval(p,x)

Evaluate the polynomial at the given point.

expr(p)

A JME expression equivalent to the given polynomial; you can substitute this into the correct answer for a "Mathematical expression" part, for example.

string(p)

A string representation of the polynomial.

latex(p)

A LaTeX representation of the polynomial.

long_division(p1,p2)

LaTeX rendering of the long division of p1 by p2.

JavaScript functions

Base object: Numbas.extensions.polynomials.Polynomial

(set it to a more convenient name, e.g. var poly = Numbas.extensions.polynomials.Polynomial)

new Polynomial(variable_name,coefficients,[modulo])

coefficients is a dictionary of degree → coefficient. If modulo is given, all coefficients will be reduced modulo that number in any calculations using this polynomial.

Polynomial.from_tree(tree,[modulo])

Create a polynomial object from a compiled JME tree

Polynomial.from_string(expr,[modulo])

Create a polynomial object from a JME string

Polynomial object methods

p.evaluate(x)

Evaluate at point x to a number

p.toLaTeX()

Render as a LaTeX string

p.isZero()

Is this polynomial zero?

p.degree()

Degree of highest power term in p with a non-zero coefficient

p.negate()

Negate every coefficient of p (returns a new polynomial)

p1.add(p2)

Add p1 to p2

p1.sub(p2)

Subtract p2 from p1

p1.mul(p2)

Mutliply p1 by p2

p.pow(n)

nth power of p

p.scale(n)

Multiply p by constant n

p.add_degree(n)

Add n to the degree of each term of p

p1.div(p2)

Divide p1 by p2. Returns an object {quotient: <polynomial>, remainder: <polynomial>}

p.mod(n)

Take each coefficient of p mod n (returns a new polynomial object)

p1.eq(p2)

Are p1 and p2 equal?

p.coefficient(d)

Coefficient of x^d in p.