In my high school algebra class I was taught that I could not factor the formula X²+1; that is, that I could not represent it as a product of two formulae as I could with the similar formula

While factoring a formula does not actually solve an equation, it is nonetheless a part of the puzzle of algebra because it provides us with one of the tools that we use in solving equations. That equation does, in fact, suggest a way to factor x²+1. We need only recall to mind the fact that multiplying two negative numbers together yields a positive number; that is we can represent any positive number as the negative of a negative number, so we have

in which . Of course that only works if we are allowed to use complex numbers.

    Suppose we restrict our factors to real numbers only. Can we still factor x²+1?

    We know that we cannot devise a pair of real-valued binomials whose product equals x²+1. Could we devise a pair of real-valued trinomials that will work? In that case we would have

We easily make the identifications A=x and C=1, so we must have

We solve that readily for and get

That particular factorization comes in handy in solving the integral .