Polynomials are defined as the sum of variables with increasing integer power and their coefficients in a certain field. For example the following might be still known from school:

P(x) = x^2 + 4x + 3

const p = new Polynomial("3x^2").add("-x^2"); // 2x^2

const p = new Polynomial("5+3x^3+6x^5").derive(2); // 120x^3+18x

Any function (see below) as well as the constructor of the Polynomial class parses its input like this.

You can pass either Objects, Doubles or Strings. Make sure strings don't contain any white-spaces or brackets. The parser doesn't analyse the string recursively.

new Polynomial({'3': 4, '5': '9'}); // 9x^5+4x^3
new Polynomial([1,2,3]); //3x^2+2x+1

new Polynomial(55); // 55x^0

new Polynomial("98x^2+4+23x^4");

The string parser passes every coefficient directly to the field parser, which allows to pass complex and rational coefficients as well:

// Example with complex numbers
Polynomial.setField("C");
new Polynomial("98x^2+i+23ix^4");

// Example with rational numbers
Polynomial.setField("Q");
new Polynomial("5/3x^3+4/3x");

Polynomial.js is held general in order to operate on various fields. Fraction.js and Complex.js build the perfect base to extend polynomials to rational and complex numbers.

• ℚ: Rational numbers supported by Fraction.js
• ℂ: Complex numbers supported by Complex.js
• ℍ: Quaternions supported by Quaternion.js
• ℤ_p: Field of integers mod p, with p prime
• ℝ: Field of real numbers

const p = new Polynomial("98x^2+4+23x^4");

console.log(p.coeff);

Polynomial.setField("Q");
Polynomial("3/2x^2-4x").mod("5x"); // 0

Polynomial.setField("Z11");
Polynomial("3x^2+x+7").gcd("3x^2+x+7"); // x^2+4x+6

Polynomial.setField("Z7");
Polynomial("9x^2+4").pow(3); // x^6+6x^4+5x^2+1

Polynomial.setField("R");
Polynomial("3x^3-1").mul(4); // 12x^3-4

// Derivative of polynomial
Polynomial.setField("Q");
Polynomial("5+3x^3+6x^5").derive(); // 30x^4+9x^2

// Integrated polynomial
Polynomial.setField("Q");
Polynomial("3x^2").integrate(); // x^3

Returns the sum of the actual polynomial and the parameter n

Returns the difference of the actual polynomial and the parameter n

Returns the product of the actual polynomial and the parameter n

Adds the product of x and y to the actual number

Returns the quotient of the actual polynomial and the parameter n

There is a global variable to enable division tracing like this, if you want to output details:

Polynomial.trace = true;
new Polynomial("x^4+3x^3+2x^2+6x")
        .div("x+3");
console.log(Polynomial.trace.map(x => x.toString())); // ["x^4+3x^3", "2x^2+6x", "0"]

Returns the negated polynomial

Returns the reciprocal polynomial

Gets the leading coefficient

Gets the leading monomial

Divide all coefficients of the polynomial by lc()

Returns the n-th derivative of the polynomial

Returns the n-th integration of the polynomial

Evaluate the polynomial at point x, using Horner's method. Type for x must be a valid value for the given field.

(Deprecated) Alias for eval.

Returns the power of the actual polynomial, raised to an integer exponent.

Returns the modulus (rest of the division) of the actual polynomial and n (this % n).

Returns the greatest common divisor of two polynomials

Returns the degree of the polynomial

Generates a string representation of the actual polynomial. This makes use of the toString() function of the field.

Generates a LaTeX representation of the actual polynomial.

Formats the actual polynomial to a Horner Scheme

Creates a copy of the actual Polynomial object

Creates a new (monic) Polynomial whose roots lie at the values provided in the array roots

Sets the field globally. Choose one of the following strings for x:

• "R": real numbers
• "Q": rational numbers
• "H": quaternions
• "C": complex numbers
• "Zp": with p being a prime number, like "Z7"
• or an object with the field operators. See examples folders for BigInt

If a really hard error occurs (parsing error, division by zero), polynomial.js throws exceptions! Please make sure you handle them correctly.

Installing Polynomial.js is as easy as cloning this repo or use one of the following command:

npm install polynomial

<script src="fraction.min.js"></script> <!-- Needed for field/ring Q -->
<script src="complex.min.js"></script> <!-- Needed for field C -->
<script src="polynomial.min.js"></script>
<script>
Polynomial.setField("C")
console.log(Polynomial("4x+3i"));
</script>

As every library I publish, Polynomial.js is also built to be as small as possible after compressing it with Google Closure Compiler in advanced mode. Thus the coding style orientates a little on maxing-out the compression rate. Please make sure you keep this style if you plan to extend the library.

After cloning the Git repository run:

npm install
npm run build

Testing the source against the shipped test suite is as easy as

npm run test

Copyright (c) 2024, Robert Eisele Licensed under the MIT license.