Polynomial Factorization

Here are some common techniques for factoring polynomials in practice.

Factoring Out Common Terms

Complete factoring

$$ ax^3 + bx^2 + cx = x (ax^2 + bx + c) $$

Partial factoring

$$ ax + ay + bx + by = a(x+y) + b(x+y) = (a+b)(x+y) $$

Square of a binomial

$$ (a+b)^2 = a^2 + 2ab + b^2 $$

Cube of a binomial

$$ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $$

Square of a trinomial

$$ (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc $$

Difference of squares

$$ a^2 - b^2 = (a+b)(a-b) $$

Sum of cubes

$$ a^3 + b^3 = (a+b)(a^2 - ab + b^2) $$

Difference of cubes

$$ a^3 - b^3 = (a-b)(a^2 + ab + b^2) $$

$$ x^2 + sx + p = (x+a)(x+b) \quad \text{where } s=a+b, \ p=ab $$

$$ ax^2 + bx + c = ax^2 + px + qx + c \quad \text{where } b=p+q, \ c=pq $$

$$ ax^4 + bx^2 + c = at^2 + bt + c \quad \text{with } t = x^2 $$

$$ P(x) = (x-x_1) \cdot Q(x) $$