Home page for accesible maths 1 1 Further Integration 1.4 The fundamental theorem of algebra 1.6 Factorizing of real polynomials

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1.5 Factors of real polynomials

A polynomial is real if all of its coefficients are real.

Lemma.

Let $f(x)$ be a real polynomial with a root $\alpha$ that is not real. Then $\bar{\alpha}$ is also a root, so $x^{2}-2(\Re\alpha)x+|\alpha|^{2}$ is a factor.

Brief proof: If $a_{n}\alpha^{n}+\ldots+a_{0}=0$ then, taking the complex conjugate,

\overline{a_{n}\alpha^{n}+\ldots+a_{0}}=a_{n}\bar{\alpha}^{n}+\ldots+a_{0}=0.

It follows that $(x-\alpha)(x-\bar{\alpha})=x^{2}-2(\Re\alpha)x+|\alpha|^{2}$ divides $f(x)$ .

Corollary.

Any real polynomial is a product of irreducible real quadratics times a product of real linear factors.