Mathematics · Ch 3 — Theory of Equations

Imaginary Roots

3.4.1

Imaginary Roots

For a quadratic with real coefficients, if $\alpha+i\beta$ is a root then $\alpha-i\beta$ is also a root — this section proves the same is true for any degree.

Note

Theorem 3.2 (Complex Conjugate Root Theorem). If a complex number $z_0$ is a root of a polynomial equation with real coefficients, then its conjugate $\overline{z_0}$ is also a root.

Proof. Let $P(x)=a_nx^n+\cdots+a_1x+a_0$ have real coefficients and $P(z_0)=0$ . Taking the conjugate of both sides, and using that conjugation respects sums/products and that $\overline{a_r}=a_r$ for each real $a_r$ :

$\overline{P(z_0)} = a_n\overline{z_0}^{\,n}+\cdots+a_1\overline{z_0}+a_0 = P(\overline{z_0}).$

Since $P(z_0)=0$ , also $\overline{P(z_0)}=\overline 0=0$ , so $P(\overline{z_0})=0$ — i.e. $\overline{z_0}$ is a root too. $\blacksquare$

Terminology note. Every real number is technically also a complex number (just as every integer is a rational number), so to single out a genuinely non-real complex number $\alpha+i\beta$ (with $\beta\ne0$ ), it is called a non-real complex number (or, by some authors, an imaginary number). The informal phrase "complex roots occur in pairs" really means: non-real roots of a real-coefficient equation occur in conjugate pairs.

A structural consequence (Remark 2). Since non-real roots of a real-coefficient equation always come in pairs, an odd-degree equation with real coefficients always has at least one real root — in fact, its number of real roots is always odd. Symmetrically, an even-degree real equation always has an even number of real roots (possibly zero). …

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