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Factoring A Trinomial (Illegal Method)

Updated on January 18, 2013

Factoring trinomials.

If I didn't lose you at that, then you're here for help, which is what I shall give (hopefully).

Factoring trinomials is easy enough, as long as, in an Ax+Bx+C trinomial, Ax=x2. But factoring out something like 4x2+29x+30 can wear on your brain. And when you have thirty of those questions in a row, and you need to get them done quickly so you can get to other homework or activities, then factoring it out mentally is a real pain. We're all in luck, though, as there is a method for factoring trinomials that can eliminate all of that nasty guessing and checking.

First of all, the leading coefficient, your A, cannot be 1. Also, if A is a negative, then you'll need to factor it out, but we'll get to that. We'll need an example trinomial first.


Okay, not too hard. You could probably do this one in your head, but who has time for that, right? Thinking is overrated anyway. So let's make it easy.

Step 1: Multiply the leading coefficient, A, by C, and form a new trinomial. Like so:


Woah. What just happened there? You took one number from a problem and multiplied it by another number in the same problem? Not on the other side of the equals sign or anything? Yeah. It's okay, I was skeptical too at first. We'll be undoing that later anyway. Besides, now we're back to our standard x2, so factoring this will be cake.

Step 2: Factor your new trinomial.


We're not done yet, though. Remember that crazy move we did earlier, you know, the one that broke mathematical law? Well, we'll undo that here.

Step 3: Undo the "illegal move" by dividing your coefficients by your old leading coefficient (A).


Wait, no, stop! Fractions in the binomials? What are you going to do now? Easy. Relax about the fractions, and remember that you are only dividing. Reduce your fractions (if you can).

Step 4: Reduce your fractions.


Now it's a little better, but you still have that pesky 5/3. We can't have that. But the good new is we can get rid of it.

Step 5: Clear any fractions by moving the denominator in front of the x.


Wait... if you solve that... just a little FOILing here and there and... no... Wow. There, above this, are your factors. To check, just multiply that, and you get:


Tada. Just a little finagling and you can get the factors of your trinomial without hurting your head. Oh, and if you're worried about that "factoring out the negative" thing I talked about earlier, don't be. Here's an example of how to factor a trinomial with a negative leading coefficient, using the illegal method.






This will work for all basic trinomials (meaning I haven't seen one that this doesn't work for, but I bet they're out there. Math is sneaky like that). Use it for yourself, tell your friends, impress your teachers, or just continue factoring in your head (I'll admit it, I do).


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      Colin Beveridge 2 years ago

      I was skeptical of it, too!

      What you're doing is:

      * multiplying the whole thing by a, so you have (1/a) [a^2 x^2 + ab x + ac]

      * replacing ax with X: (1/a)[X^2 + bX + ac]

      * factorising: (1/a)(X - stuff)(X - stuff)

      * returning to x: (1/a)(ax - stuff)(ax - stuff)

      * and dividing the a out in the appropriate place.