Factoring the Illegal Way

Remember that the standard form of a trinomial is .

Example:

This is difficult to factor because the coefficient of the squared term (a) is not 1. Therefore I remove the 6 by multiplying it with the c term (-3). My new trinomial is

Now this trinomial is easily factored into .

I did an “illegal move,” and I now need to “undo” it. Since I multiplied by 6 in the first step, to “undo” it, I now divide each constant by 6.

I now have a factored form with fractions. That is not acceptable so I first reduce the fractions to lowest terms.

The binomials still have fractions that cannot be reduced, so I simply take the denominator of the fraction and squeeze it in front of the x in the binomial, making the denominator the coefficient of x.

It works every time!

The only problem I have had with students using this method is they forget to undo the illegal move. You must do both steps!!!

This does not work if a is a negative number. Factor out a -1 and then proceed.

Here’s another example:

Because a is not 1, I perform the illegal move. Multiply by 2.

Factor using the sign clues.

Now I must undo the illegal move. Divide by 2.

Simplify the fractions if I can.

Since I still have a fraction, I move the denominator of the fraction in front of the variable.

One More Example:

Perform the illegal move. Multiply by 12.

Factor using the sign clues.

Undo the illegal move. Divide by 12.

Reduce fractions.

Remove fractions.