Factoring polynomials and trinomials is one of the most important topics in basic algebra. There is no single, universal method to factor all polynomials; instead, there are a handful of techniques that apply to specific types of polynomials. If you recognize which types of polynomials are best solved by each technique, it will make factoring simpler and more intuitive.
The Guess and Check Method
Trinomials are divided into two types: monic and nonmonic. If the leading coefficient of a trinomial (the number attached to the x^2 term) is 1, then the trinomial is monic. These are the easiest polynomials to factor using the guess and check method. Write the two factors in the form (x )(x ). After the x term in both factors will be a number. The numbers are those that multiply to make the constant and add to make the middle coefficient. For example, to find the factors of the monic trinomial x^2 - 4x + 3, find the pair of numbers that multiply to make 3 and add to make -4. These numbers are -1 and -3, because -1 x -3 = 3 and -1 + -3 = -4. The factored form of the trinomial is therefore (x - 1)(x - 3).
The AC Method
Nonmonic trinomials are generally more difficult to factor. Use a modified form of the guess and check method to take into account the fact that the coefficient is not 1. The method is called the AC method because instead of finding the pair of numbers that multiply to make the constant, you have to find a pair that multiplies to make AC, the product of the leading coefficient and the constant. For example, given the polynomial 2x^2 -7x + 6, use the AC method to find the pair of numbers that multiply to make the product of 2 and 6 (12) and add to make -7. These two numbers are -3 and -4. Once you have found the numbers, split the middle term into two terms with those coefficients and then factor by grouping. Split the middle term in the polynomial 2x^2 - 7x + 6 to make 2x^2 - 4x - 3x + 6, then factor by grouping.
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Factoring by Grouping
The method most often used to factor polynomials with more than three terms is the grouping method. The polynomial is split into two groups, which are then factored independently. The goal is to extract a factor so that the paired factor is the same for both groups. This factor is then extracted from the entire polynomial to get it to factored form. For example, split the polynomial 2x^2 - 4x - 3x + 6 into two groups, 2x^2 - 4x and -3x + 6. Extract the common factor from both groups to get 2x(x - 2) and -3(x - 2). The groups share a paired factor (x - 2), which can be extracted to make the polynomial 2x(x - 2) - 3(x - 2) equal to (x - 2)(2x - 3). If your paired factors aren't equal after extracting a common factor, extract a different factor from one of the groups, or group the terms a different way.
Sum and Difference Formulas
The sum and difference of cubes formula and the difference of squares formula are the key to factoring binomials, which are polynomials with only two terms. The sum of cubes formula is a^3 + b^3 = (a + b)(a^2 - ab + b^2), while the difference of cubes formula is only slightly different: a^3 - b^3 = (a - b)(a^2 + ab + b^2). The difference of squares formula is a^2 - b^2 = (a + b)(a - b). In all three formulas, "a" and "b" can be either variables or constants. For example, to factor the binomial x^3 - 27, make a = x^3 and b = 27 and find the value of a, b, a^2, b^2. Plug these values into the formula to get the factored form (x - 3)(x^2 + 3x + 9).