Equation Solving

Solve a polynomial equation:

solve x^2 + 4x + 6 = 0

Solve a system of linear equations:

x+y=10, x-y=4

Solve an equation with parameters:

solve a x^2 + b x + c = 0 for x

Polynomials

Compute properties of a polynomial in several variables:

x^3 + x^2 y + x y^2 + y^3

Factor a polynomial:

factor 2x^5 - 19x^4 + 58x^3 - 67x^2 + 56x - 48

Rational Functions

Compute properties of a rational function:

(x^2-1)/(x^2+1)

Compute a partial fraction decomposition:

partial fractions (x^2-4)/(x^4-x)

Simplification

Simplify an expression:

1/(1+sqrt(2))

simplify x^5-20x^4+163x^3-676x^2+1424x-1209

simplify cos(arcsin(x)/2)

Vectors

Compute properties of a vector:

vector {2, -5, 4}

Compute a cross product:

{1/4, -1/2, 1} cross {1/3, 1, -2/3}

Matrices & Linear Algebra

Do basic arithmetic on matrices:

{{0,-1},{1,0}}.{{1,2},{3,4}}+{{2,-1},{-1,2}}

Compute eigenvalues and eigenvectors of a matrix:

eigenvalues {{4,1},{2,-1}}

Quaternions

Get information about a quaternion:

quaternion: 0+2i-j-3k

Do calculations with quaternions:

quaternion -Sin[Pi]+3i+4j+3k multiplied by -1j+3.9i+4-3k

Finite Groups

Get information about a finite group:

alternating group A_5

Ask about a property of a group:

order of the monster group

Do algebra with permutations:

perm (1 2 3 4)^3(1 2 3)^-1

Finite Fields

Compute properties of a finite field:

Z/7Z

Compute a specific property:

number of primitive polynomials of GF(3125)

Domain & Range

Compute the domain of a function:

domain of f(x) = x/(x^2-1)

Compute the range of a function:

range of e^(-x^2)