Factoring

Package: SYMBOLIC_PACKAGE. Registered by createEngine(); for a slimmer engine, register it explicitly (see choosing packages).

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.

Pi, e and the other irrational constants

Pi is irrational: no fraction equals it, so the engine holds it as the sixteen-digit decimal nearest to it, and x^2 - pi is a polynomial over that long fraction. A quadratic is decided without any searching: it has a rational root exactly when the number under the square root in the quadratic formula (its discriminant, b^2 - 4ac) is the square of a fraction. For x^2 - pi it is 4pi, which is not, so the answer is the polynomial as written, as it is for x^2 - 2. A shared variable still comes out in front.

factor(x^2 - pi) // x^2-3.1415926536
factor(x^2 - e) // x^2-2.7182818285
factor(x^2 + pi*x) // x*(x+3.1415926536)
factor((x - pi)^2) // (x-3.1415926536)^2

A cubic or anything higher has no such shortcut. Finding its rational roots means trying every fraction whose top divides the last coefficient and whose bottom divides the first, and the divisors of pi’s sixteen-digit fraction are too many to list, so it is refused rather than guessed at. Leaving it whole would claim it has no rational factor, which was never checked. solve finds its roots as decimals instead.

factor(x^3 - pi) // ERROR: This polynomial cannot be factored: a number in it is too long as a fraction (as pi and e are) to try every fraction that could be a root. solve finds its roots as decimals.
solve(x^3 = pi, x) // [-0.7322959438-1.2683737808i, -0.7322959438+1.2683737808i, 1.46]

Pi squared, typed as pi^2, is rounded to a double of its own, which is not exactly the square of pi’s double, so x^2 - 2*pi*x + pi^2 stays as written where (x - pi)^2, which is expanded exactly, factors back.

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