Were the real irreducible polynomials understood before complex numbers were developed

Question

Thanks to the algebraic closure of the complex numbers, it is easy to verify that the only irreducible real polynomials are of the form $(x+z)(x+\bar{z})$ for $z \in \mathbb{C}$ with $Im(z)\neq0$. Was this understood before the advent of complex numbers?

That is, was it understood that all real polynomials can be broken down into linear and quadratic factors before complex numbers were first used, or was the process of factorization of polynomials and the key results therein a more recent development?

Answer 1

The Babylonians basically knew how to solve quadratic equations, so they must have had a concept of "unsolvable" or "undetermined" according to J.Dieudonné (Geschichte der Mathematik, 1700-1900, p.57) He mentions further that Vieta's formulas have been known in the 17th century, so they simply did not think about proving the FTA, which started in the middle of the 18th century. Complex numbers have definitely been used by Gauß, so they were later than FTA, and later than the Babylonians.

Answer 2

What do you mean "understood"? This fact was not proved before complex numbers were known. It may have been expected from work on integrating rational functions (to do that you want to factor the denominator as much as possible), but the idea of proving it at that time seems unrealistic.

The fact you ask about is equivalent to the complex numbers being algebraically closed. Here's why. All real quadratic polynomials factor over $\mathbf C$, and if $f(z)$ is a nonconstant polynomial with complex coefficients then $f(z)\overline{f}(z)$ is a nonconstant polynomial with real coefficients, so if we know that irreducible real polynomials have degree $1$ or $2$ then $f(z)\overline{f}(z)$ would have a root $r$ in $\mathbf C$, so $f(r) = 0$ or $\overline{f}(r) = 0$. The second condition implies $f(\overline{r}) = 0$, so either $r$ or $\overline{r}$ is a root of $f$ and thus all nonconstant complex polynomials have a complex root.