Exercise · ZW-FUNDAMENTAL-THEOREM-OF-ALGEBRA-001 · ★☆☆ Basic
How many complex roots, counted with multiplicity, does \(z^5-3z+1\) have?
Choose one option.
Show answer
- C. \(5\)
Explanation
The theorem guarantees as many roots as the degree, counted with multiplicity: \(5\).
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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- ZW-FUNDAMENTAL-THEOREM-OF-ALGEBRA-007 Which of these statements about polynomials and complex roots is true?
- ZW-FUNDAMENTAL-THEOREM-OF-ALGEBRA-008 The monic polynomial with roots 1 (double) and -2 is:
- ZW-FUNDAMENTAL-THEOREM-OF-ALGEBRA-009 In the proof via Liouville, if p had no roots, which function would be entire and bounded?