# Polynomials The 17 definitions of the polynomials library, each with its Epsil spelling, its MathJSON name, its signature and its full description. Each definition is listed under its Epsil spelling (the MathJSON name when it has none), with its signature in the engine's type syntax. The [Standard Library](/library/) page is the one-page index of every category. ## Definitions ### apart MathJSON `Apart` · `(value, symbol?) -> value` Alias for PartialFraction. Decompose a rational expression into partial fractions. ### cancel MathJSON `Cancel` · `(value, symbol?) -> value` Cancel common polynomial factors in the numerator and denominator of a rational expression. Example: Cancel((x² - 1)/(x - 1), x) → x + 1 ### coefficientList MathJSON `CoefficientList` · `(value, symbol?) -> list` Return the list of coefficients of a polynomial, from highest to lowest degree. Example: CoefficientList(x³ + 2x + 1, x) → [1, 0, 2, 1] ### discriminant MathJSON `Discriminant` · `(value, symbol?) -> value` Return the discriminant of a polynomial. Example: Discriminant(x² - 5x + 6, x) → 1 ### distribute MathJSON `Distribute` · `(value) -> value` Distribute multiplication over addition ### expand MathJSON `Expand` · `(value) -> value` Expand out products and positive integer powers ### expandAll MathJSON `ExpandAll` · `(value) -> value` Recursively expand out products and positive integer powers ### factor MathJSON `Factor` · `(value, symbol?) -> value` Factor a polynomial expression into a product of irreducible factors. Supports perfect square trinomials, difference of squares, and quadratic factoring with rational roots. Example: Factor(x² + 5x + 6) → (x+2)(x+3), Factor(x² + 2x + 1) → (x+1)² ### partialFraction MathJSON `PartialFraction` · `(value, symbol?) -> value` Decompose a rational expression into partial fractions. Example: PartialFraction(1/((x+1)(x+2)), x) → 1/(x+1) - 1/(x+2) ### polynomial MathJSON `Polynomial` · `(list, symbol) -> value` Construct a polynomial from a list of coefficients (highest to lowest degree) and a variable. Example: Polynomial([1, 0, 2, 1], x) → x³ + 2x + 1 ### polynomialDegree MathJSON `PolynomialDegree` · `(value, symbol?) -> integer` Return the degree of a polynomial with respect to a variable. Example: PolynomialDegree(x³ + 2x + 1, x) → 3 ### polynomialGCD MathJSON `PolynomialGCD` · `(a: value, b: value, variable: symbol?) -> value` Return the greatest common divisor of two polynomials. Example: PolynomialGCD(x² - 1, x - 1, x) → x - 1 ### polynomialQuotient MathJSON `PolynomialQuotient` · `(dividend: value, divisor: value, variable: symbol?) -> value` Return the quotient of polynomial division of dividend by divisor. Example: PolynomialQuotient(x³ - 1, x - 1, x) → x² + x + 1 ### polynomialRemainder MathJSON `PolynomialRemainder` · `(dividend: value, divisor: value, variable: symbol?) -> value` Return the remainder of polynomial division of dividend by divisor. Example: PolynomialRemainder(x³ + 2x + 1, x + 1, x) → -2 ### polynomialRoots MathJSON `PolynomialRoots` · `(value, symbol?) -> set` Return the roots of a polynomial expression. Example: PolynomialRoots(x² - 5x + 6, x) → {2, 3} ### resultant MathJSON `Resultant` · `(a: value, b: value, variable: symbol?) -> value` Return the resultant of two polynomials with respect to a variable. It is zero iff the polynomials share a common factor. Example: Resultant(x² - 1, x - 1, x) → 0 ### together MathJSON `Together` · `(value) -> value` Combine rational expressions into a single fraction