AC Method
The AC method is an algorithm for factoring quadratic polynomials of the form p(x)=Ax^2+Bx+C with integer coefficients. As its name suggests, the crux of the algorithm is to consider the multiplicative factors of the product of the coefficients A and C. More precisely, the goal is to find an integer pair s and t satisfying AC=st and B=s+t simultaneously, whereby one can rewrite p(x) in the form
| p(x)=Ax^2+(s+t)x+C, |
(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 p(x)=4x^2-12x-7 having coefficients A=4, B=-12, and C=-7. To begin the AC factorization, consider the product AC=4×-7=-28. By observation, -28=-14×2 while -12=-14+2; in particular, this guarantees that p can be rewritten so that
| p(x)=4x^2-14x+2x-7. |
(2)
|
This four-term expression for p can be factored by grouping:
| p(x)=2x(2x-7)+(2x-7), |
(3)
|
and so
| p(x)=(2x-7)(2x+1). |
(4)
|
One can easily see that the above method generalizes to certain polynomials of the form q(x)=Ax^(2n)+Bx^n+C for positive integers n>=1, though the result will be a factorization into pairs of polynomials of degree n 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 s and t hinges on observation and/or guess-and-check; this can be particularly problematic when the product AC has a large number of factors. Moreover, while the quadratic formula illustrates immediately the existence of irrational and/or imaginary roots, the AC method often disguises such behavior and thus requires a degree of "pre-processing," e.g., by analyzing the polynomial discriminant.
See also
Berlekamp-Zassenhaus Algorithm, Factor, Factorization, Fundamental Theorem of Algebra, Kronecker's Algorithm, Polynomial Discriminant, Polynomial Factor Theorem, Polynomial Roots, Prime FactorizationThis entry contributed by Christopher Stover
Explore with Wolfram|Alpha
More things to try:
Cite this as:
Stover, Christopher. "AC Method." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ACMethod.html