Why do complex roots of a polynomial with real coefficients occur in conjugate pairs?

In the study of polynomial equations with real coefficients, complex roots must occur in conjugate pairs because of the way these polynomials are structured. As recognized by Leonhard Euler in 1742, if a complex number of the form $a+bi$ (where $i=\sqrt{-1}$) is a root of a polynomial $f(x)=0$, then its conjugate, $a-bi$, must also be a root.1

This property arises because the coefficients of the polynomial are real numbers. When you evaluate the polynomial at a complex number and its conjugate, the imaginary components cancel out, ensuring the equation remains consistent with its real-valued coefficients. Mathematically, this implies that any polynomial with real coefficients can be decomposed into a product of linear and quadratic factors. This result is significant in mathematics, particularly in the theory of integration, as it allows for the decomposition of rational functions into simpler, integrable parts.1