The AC Method is a technique used to factorize polynomials in a way that expresses them as a product of factors that cannot be further simplified. By using this method, it is possible to factorize polynomials of a higher degree into irreducible factors of a lower degree. This technique can come in handy in various mathematical applications, such as solving quadratic equations in the form \(ax^2+bx+c\) without having to use the quadratic formula.

The method involves a few simple steps that can be followed to factorize a given polynomial into irreducible factors. The process may seem complicated, but it is simple and straightforward.

  • Given a quadratic expression of the form \(ax^2+bx+c\), find all the factors of the product of the coefficients \(a \cdot c\).

  • If \(a \cdot c \gt 0\), add the factors to form the coefficient \(b\) or if \(a \cdot c \lt 0\), subtract the factors to form the coefficient \(b\).

  • Replace the middle term \(bx\) with the new terms, group the equation and find the common factors in each group of factors.

Example

The polynomial \(2x^2+7x+3\) can be factorised in the form \((x+3)(2x+1)\). We have to find two numbers, \(x_1,x_2\), whose product \(P\) is \(6\) and the sum \(S\) is \(7\).


We can use this simple scheme to find the numbers that satisfy our constraints. \begin{array}{rrrr} & x_1 & x_2 & P & S \\ \hline & 1 & 6 & 6 & 7 \\ & -1 & -6 & 6 & -7 \\ & 2 & 3 & 6 & 5 \\ & -2 & -3 & 6 & -6 \\ \end{array}


The first row \(x_1=1, x_2 = 6\) satisfies our constraints. Now replace the term \(bx\) as the sum of the values found of \(x_1, x_2\). The polynomial becomes: \[2x^2 +7x +3 = 2x^2 + x +6x +3 = 0\]


The terms can be grouped as follows: \[ \begin{align} &2x^2 + x = x(2x +1)\\[0.5em] &6x +3 = 3(2x +1) \\[0.5em] \end{align} \]


We get: \[ 2x^2 + x +6x +3 = x(2x +1) + 3(2x +1) \]


By finding the common terms in each group, we obtain that the polynomial can be decomposed as: \[ (x +3)(2x +1)\]

The polynomial can be factorised in the form: \[ (x +3)(2x +1)\]

Particular products of powers, binomials, and trinomials are known as notable products. They are particularly important for simplifying algebraic expressions, factorising polynomials, and solving mathematical problems.