Methods for Factoring Trinomials

Factoring trinomials is one of the most difficult concepts in algebra.
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If there is one math subject nearly every student finds challenging when he or she first encounters it, it is algebra, particularly the factoring of trinomials. There are several methods for factoring trinomials, and none of them are what anyone would call "easy." However, each can be understood with consistent study and practice.

What Is a Trinomial?

First, you must know what a polynomial is. A polynomial is an algebraic equation that has terms, combinations of numbers and variables like 3x and 5y. Some examples of polynomials are 2x + 3, 3xy - 4y and 3x + 4xy - 5y. That last example is called a trinomial. A trinomial is a polynomial with three terms.

Greatest Common Factor

The first, and arguably "easiest," method for factoring trinomials is by finding the greatest common factor -- the largest number, variable or term the three terms have in common. For example, with the trinomial 2x^2 + 6x + 4, the number 2 is the only number all three terms have in common, so when you factor out 2, you get 2(x^2 + 3x + 2). The trinomial inside of the parentheses can actually be factored further.

Factoring Quadratic Trinomials

The trinomial x^2 + 3x + 2 is a quadratic trinomial because it has a term with a power of two. To factor this polynomial, you must know some rules about quadratics. First, the factors of quadratic trinomials are usually two binomials, such as x + 2 or 2y - 3. Second, the first term of the quadratic trinomial is the product of the first terms of the two binomials. Third, the last term of the quadratic trinomial is the product of the last terms of the two binomials. Fourth, the coefficient of the middle term of the quadratic trinomial is the sum of the last terms of the two binomials. Fifth, if all the signs in the quadratic trinomial are positive, all signs in both binomials are positive.

Factoring Example

To factor the quadratic trinomial x^2 + 3x + 2, start with two sets of parentheses, ( )( ). Do the second step by writing an x in both parentheses, (x )(x ). The variable x^2 equals x multiplied by x, fulfilling the first rule. The third step states the last term of the trinomial is the product of the last terms of both binomials, so the last must be either 1 and 2 or -1 and -2 -- both of these equal 2. The fourth step states the middle term coefficient is the sum of the last terms of the two binomials. Only 1 and 2 equals 3, so the solution is (x + 1)(x + 2). Also, the fifth rule is satisfied as well.

Special Cases and Other Information

Sometimes you may have to rewrite the trinomial to make factoring easier. The trinomial 3x + 2y + 3xy is easier to solve in the more logical order of 3x + 3xy + 2y, with all of the like terms together. Rearranging the order of trinomials can only be used if all of the signs in the trinomial are positive. Also, some trinomials cannot be factored, such as x^2 + 4x +2. There is no way this trinomial can be broken down any further.