Overview
Factoring polynomial expressions is similar to factoring special polynomials. The difference is that both elements do not have to be perfect squares. It is the reverse of multiplying two binomials by using FOIL.
Factoring Trinomials
Trinomials of the form x2 +bx +c, when c>0 have certain aspects in common. The coefficient of the squared term is 1, so the only form to factor is the variable itself, in this case x times x. The constant c is a positive value. If the constant c is a positive number, but not a perfect square, it may have a variety of factors.
Using FOIL in Reverse
In order to multiply two binomials, FOIL is used, for the first term, the outer terms, the inner terms, and the last terms. It is also useful when factoring trinomials, only it is used to unravel the trinomial. In these examples, the first term, x2, can be factored as x ·x. The factorization will have the form of (x + ___) (x + ____), so that first term is already known in these examples.
The Constant Term
The constant term c has a fixed numeric value, so that the last terms in each binomial will be factors of c. Suppose the binomial is x2 +7x +12. The constant 12 has a number of factors, as 12 ·1 is 12, 6·2 is 12, and 3·4 is also 12. The constant 12 also has negative factors, as -12·-1 is 12, ·-6·-2 is 12, and -3·-4 is also 12.
The Middle Terms
Since there are usually a number of factors for the constant, there has to be a way to narrow them down and choose the correct one. Recall that the inner terms and the outer terms are added, while the last terms are multiplied. In order for a pair of factors to work, they must be added so that their sum is the coefficient of the middle term of the trinomial. In this example, the coefficient of the middle term is 7. The first pair of factors, 12 +1, equals 13; the second pair, 6 +2, equals 8; and the third pair, 3 +4, equals 7. By substitution, x2 +7x + 12 factors as (x +3) (x +4).
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