Factoring

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Factoring is the reverse of expanding: it takes a polynomial written out as a sum, like x^2+3x+2, and rewrites it as a product of simpler pieces, like (x+1)*(x+2). It is how you find the values that make an expression zero, and how a fraction of polynomials reveals what cancels. Like the other algebra forms, this does not need a trailing arrow.

factor is the inverse: it writes a polynomial as a product.

factor(x^2-4) // (x-2)*(x+2)
factor(x^2+3x+2) // (x+1)*(x+2)
factor(x^2-2x+1) // (x-1)^2
factor(2x^2+4x) // 2x*(x+2)
factor(x^3-1) // (x-1)*(x^2+x+1)

A repeated root becomes a power rather than a repeated factor, and a shared constant or variable comes out in front.

Factoring over what

Factoring only means something once you say over which numbers. x^2-2 factors over the real numbers as (x-sqrt(2))(x+sqrt(2)), and x^2+1 factors only over the complex numbers. Both are left alone here, because this factors over the rationals.

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

That is an answer rather than a failure. A polynomial with no rational roots is returned as-is, including in the cases where it would split into higher-degree rational pieces, which are not searched for.

More than one variable

Factoring in several variables at once is a much harder problem than in one, so this recognises the standard shapes rather than running a general algorithm.

factor(x^2-y^2) // (x-y)*(x+y)
factor(x^3-8y^3) // (x-2y)*(x^2+2x*y+4y^2)
factor(x^2+2x*y+y^2) // (x+y)^2
factor(a*x+a*y+b*x+b*y) // (a+b)*(x+y)

A difference of squares, a sum or difference of cubes, a perfect-square trinomial, and four terms that group into two pairs. Anything else stops after any shared constant and variable have been taken out.

factor(x^2+3x*y+y^2) // x^2+3x*y+y^2