AC Method

The  method is an algorithm for factoring quadratic polynomials of the form  with integercoefficients. As its name suggests, the crux of the algorithm is to consider the multiplicative factors of the product of the coefficients  and . More precisely, the goal is to find an integer pair  and  satisfying  and simultaneously, whereby one can rewrite  in the form

(1)

and to factor the remaining four-term polynomial by grouping into a product of linear factors with integer coefficients.

For example, consider the polynomial  having coefficients , , and . To begin the  factorization, consider the product . By observation,  while ; in particular, this guarantees that  can be rewritten so that

(2)

This four-term expression for  can be factored by grouping:

(3)

and so

(4)

One can easily see that the above method generalizes to certain polynomials of the form  for positive integers , though the result will be a factorization into pairs of polynomials of degree  which aren't necessarily linear.

This procedure is an alternative to the more straightforward utilization of the quadratic formula and has a number of drawbacks. For example, finding  and  hinges on observation and/or guess-and-check; this can be particularly problematic when the product  has a large number of factors. Moreover, while the quadratic formula illustrates immediately the existence of irrational and/or imaginary roots, the  method often disguises such behavior and thus requires a degree of "pre-processing," e.g., by analyzing the polynomial discriminant.