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.

Factor Trinomials with Leading Coefficient Equal to 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


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.


About jrh794

I am a sixty-five year old math instructor at Southern Oregon University. I taught at the College of the Siskiyous in Weed California for twenty-six years. Prior to that I worked as a computer programmer, carpenter and in various other jobs. I graduated from Rice University in 1967 and have a MS in Operations Research from Stanford. In the past I have hand-built a stone house and taken long solo bicycle tours. Now I ride my mountain bike and play golf for recreation.
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15 Responses to The Best Way to Factor Trinomials

  1. Anonymous says:

    This helped me SO much!!!

  2. Anonymous says:

    Wow, it made it so much easier!

  3. Anonymous says:

    You can multiply the coefficient outside the bracket to get rid of the fraction in one of the brackets. This method is very efficient!

  4. Anonymous says:

    ots very good, but my teacher said it to me before that

  5. how do I solve this?
    Can you give me a steps??

    • Theresa says:

      when it comes to x^2 + bx + c = 0
      factors of C would have to had up to B
      For example:
      x^2-17x+30 = 0
      Find Factors of 30 = (5,6) (1,30) (2,15) (3,10)
      Now which set would equal to ___ + ___ = 17 ?
      2 + 15 = 17
      Then you put them into your equation (x + __) (x + __)

  6. Bill says:

    x^2 – 17x + 30
    what are the factors of thirty
    1, 30
    2, 15
    3, 10
    5, 6
    And which ones can be used to get 17. 2 and 15.
    Both must be negative to get +30 and -17 so the numbers are -2 and -15
    thus the factored version is (x-2)(x-15) and so x is 2 and 15

  7. Thank you so much. This really helped me, A LOT. 🙂

  8. Anonymous says:

    I love it!!! So amazed at how quick it is. =)

  9. christina says:

    Brilliant. You are brilliant. Thank you so much for posting the easiest method I’ve found on the Internet on factoring trinomials with a leading coefficient that isn’t zero.

  10. Gina Donahue says:

    Oh dear god please teach it to them. Even if just as an alternative. Factoring never fails to make me cry; I am 36 and back in school after 15 years– after failing out of math twice. Now I’m getting 100s… except when it comes to factoring trinomials! We’re not even supposed to use the quadratic formula for equations because it’s ‘too hard’– but guessing? Literal trial and error GUESSING? Especially when one has discalculia and transposes the damned multiplication table? No no no no– tell them the expected way and the easy way, and then how to make their methods look like the hard way once they have the answer– or like I had, they’ll just get discouraged and think they’re dumb in math. 😦

  11. Unknown says:

    Do you know WHY it works? I am researching it for a essay, and I need to know why.

  12. MJP says:

    Because when you ignore the leading coefficient of the polynomial and just create a product of 2 binomials you need to multiply the leading coefficient back into the polynomial. It affects both constant terms of the binomials and must be divided from each…

  13. Alberta says:

    I love the method but I don’t understand why the leading coefficient should be attached to the factorized terms after..can we do away with it ?

  14. Bret Johns says:

    Except there is a mistake in the next to the last problem The equation is 12x^2 +13x+3=0 when I believe it should be 12x^2+13x+36=0. It does ask for factors of 36.

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