Factoring X^{2}+1

In my high school algebra class I was taught
that I could not factor the formula X^{2}+1; that is, that I could not
represent it as a product of two formulae as I could with the similar formula

(Eq'n 1)

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 x^{2}+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

(Eq'n 2)

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 x^{2}+1?

We know that we cannot devise a pair of
real-valued binomials whose product equals x^{2}+1. Could we devise a
pair of real-valued trinomials that will work? In that case we would have

(Eq'n 3)

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

(Eq'n 4)

We solve that readily for and get

(Eq'n 5)

That particular factorization comes in handy in solving the integral .

efefhghgefef