Eleventh grader Tiara asks the following question:

"How do you factor -1 + p²?"

Well Tiara, I'm going to give you two answers to that question. The first one is the answer you're probably looking for. It's simple, it's straightforward, and it's uncomplicated.

Then I'll give you an answer that no one would ever use, because it's overcomplicated - like using a sledgehammer to kill a mosquito. But I'm going to give it to you anyway, just because being overcomplicated amuses me.

Fly Swatter Approach

Reverse the order of the terms. Now instead of -1 + p², you have p² - 1, which is a Difference of Squares. Therefore, it factors into:

(p - 1)(p + 1)

Sledgehammer Approach

My ridiculous sledgehammer approach requires that you have a rudimentary understanding of i, the imaginary unit. If you don't understand that, all you really need to know is that i is used to represent the square root of -1.

If you include the imaginary unit in your thinking, then you can rewrite your expression like this:

-1 + p² = i² - p²i² 

And now you've got a difference of squares, so it factors like this:

(i - pi)(i + pi)

Each of these factors can have an i factored out, giving us:

i²(1 - p)(1 + p)
-1(1 - p)(1 + p)

Distributing that -1 through the first factor gives us (-1 + p)(1 + p), and rearranging the terms of the first factor gives us:

(p - 1)(p + 1), which is the same as the answer we obtained with the fly swatter method.

Who would ever want to use the second approach? No one. Except ridiculous math teachers who think it's funny to take simple problems and make them absurdly complex.

Or bloggers who think their blog post would have been too short otherwise.

Professor Puzzler