## Are We Lying About Factoring?

Posted by mark schwartz on May 9, 2016

This is a short piece. I didn’t mean to do this but I was playing with factoring and here’s the thought I had.

Using a simple example, the factors of x^{2} + 2x + 1 are (x + 1)(x + 1). However, what is also a set of factors of this trinomial is ( ̶ x ̶ 1) ( ̶ x ̶ 1). Although this latter is technically correct, it is not traditionally correct. If a student in class gave you this as an answer, would you accept it? Technically, yes; traditionally, no. You know what a computer-based delivery system would say … and that’s because it’s been programmed by traditionalists.

So, why won’t we accept ( ̶ x ̶ 1) ( ̶ x ̶ 1) as a pair of factors? We teach factoring as a prelude to solving an equation. The expression x^{2} + 2x + 1 becomes an equation by simply stating that the expression equals something, like 5 or y. In solving the equation using the factoring method, would it make a difference?

Given the equation x^{2} + 4x + 1 = – 3, adding 3 to both sides gives x^{2} + 4x + 4 = 0. Solving this by factoring we get (x + 2)(x + 2) = 0 and setting both to zero, x equals – 2. If we now set the factors to be (-x – 2)(-x – 2) and set both to zero, x again equals – 2. So far, it seems to work.

When wouldn’t it work? I don’t know. Just a thought to share with you …

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