Themes > Science > Mathematics > About Mathematics, Generalities > Fatorials & Factoring > Factoring Polynomials

Given a complicated algebraic expression, it is often useful to factor it into the product of several simpler terms. For example, 2x3 + 8x2y can be factored as 2x2(x + 4y). Determining the factors of a given polynomial may be a simple matter of inspection or may require trial and error. Not all polynomials, however, can be factored, and these are called prime polynomials.

Some common factorizations are given in the following examples.

Trinomials of the form:

x2 + 2xy + y2 = (x + y)(x + y) = (x + y)2

x2 - 2xy + y2 = (x - y)(x - y) = (x - y)2

25x2 + 20xy + 4y2 = (5x + 2y)2

The difference of two squares:

x2 - y2 = (x + y)(x - y)

25x2 - 16y2 = (5x + 4y)(5x - 4y)

Trinomials of the form:

x2 + (a + b)x + ab = (x + a)(x + b)

x2 + 7x + 10 = (x + 5)(x + 2)

Sums and differences of cubes:

x3 ± y3 = (x ± y)(x2 ± xy + y2)

x3 + 8y3 = (x + 2y)(x2 - 2xy + 4y2)

Grouping may often be useful in factoring; terms that are similar are grouped wherever possible, as in the following example:

2x2z + x2y - 6xz - 3xy = x2(2z + y) - 3x(2z + y) = (x2 - 3x)(2z + y) = x(x - 3)(2z + y)