Expand a polynomial as a simple sum of terms:

Expand a product of polynomials:

Expand a polynomial of several variables:

Calculate the characteristic polynomial of a diagonal matrix:

The characteristic polynomial is the determinant of the following matrix:

Expand the product of diagonal elements to get the characteristic polynomial:

This can be directly computed using the CharacteristicPolynomial function:

Cyclotomic polynomials are monic, with integer coefficients, and are irreducible over the rational numbers.

The 16th cyclotomic polynomial is given below:

Use Expand to show that it has integer coefficients:

These polynomials can be directly computed using the Cyclotomic function:

Expand can be used to verify that two expressions are equal:

Many functions give results in unexpanded form:

Apply Expand:

Apply Expand to a polynomial:

Factor is essentially the inverse of Expand:

When no powers are involved, Distribute gives the same results as Expand:

Direct application of the distributive law often generates far more terms than are needed:

Sum them:

Use NonCommutativeExpand to expand noncommutative polynomials:

Specify names for addition and multiplication operations:

Stylize a binomial expansion involving two emojis:

Expand a binomial raised to a large power:

Create a nested pattern corresponding to an additive cellular automaton (rule 60):