## The Best Way to Factor Trinomials

Warning to all beginning algebra students:

The answers you get using this method will not be the same as those in your text nor what your math instructor expects to see, but they will be correct and more useful.  You will also get them faster.

To factor any trinomial, just fill in the blanks in this question. (We are looking for factors containing only rational numbers.)

The Factoring Question

Here is how the method works with trinomials when the coefficient of $x^2$  is one.

As you can see, answering the question may get tedious, but the method is simple, efficient and works with any combination of plus and minus signs. But so far it is not much different than what you see in an algebra text.  What is different is factoring trinomials when the coefficient of $x^2$  is not equal to one.  Here is how it goes.

Factor Trinomials with Leading Coefficient Not Equal to One.

Now you can see that the method is faster then guessing and checking or factoring by grouping, the typical methods found in modern elementary algebra texts.  Note that the method with trinomials when the coefficient of $x^2$  is one is just a special case of factoring trinomials when the coefficient of $x^2$  is not equal to one.  Here is an explanatory example.

Factor Trinomials with Leading Coefficient Equal to One as Special Case.

This method is “best” because it is more efficient than traditional techniques and because it works with all trinomials so no distinguishing cases are needed.  It is also best because the resulting factored form is more useful.  For instance, once a quadratic equation is put into its “new” factored form, the answers can be found by inspection.  Here is an example.

Solve a Quadratic Equation by Factoring

The factored form makes some partial fraction problems easier to work as in this example.

Partial Fraction Decomposition

There are other examples in this powerpoint presentation that I gave in the fall of 2010 to our Mathematical Perspectives class.  Here is one of the explanations for why the method works that I gave to that class.

Why the Method Works

Note:

This is the best way I know to factor trinomials but I don’t teach it to my algebra students.  The reason is that they will be at a disadvantage in subsequent math classes since their texts and their instructors will expect  answers in the traditional form.