Exercise · ZW-FUNDAMENTAL-THEOREM-OF-ALGEBRA-007 · ★★☆ Intermediate

Which of these statements about polynomials and complex roots is true?

Choose one option.

Show answer
  • D. Every real polynomial of degree \(n\) has \(n\) complex roots with multiplicity

Explanation

That is exactly the theorem. \(z^2+1\) has \(\pm i\), \(\mathbb{R}\) is not closed (\(x^2+1\)) and non-real roots do come in conjugate pairs.

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Review the concept: Fundamental theorem of algebra

Every non-constant polynomial \(p\) with complex coefficients, of degree \(n\), has at least one root in \(\mathbb{C}\); hence it splits into \(n\) linear factors \(z-z_k\), with leading coefficient \(a_n\).

\[p(z)=a_n\prod_{k=1}^{n}(z-z_k)\]

More practice problems: fundamental theorem of algebra

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