Exercise · ZW-FUNDAMENTAL-THEOREM-OF-ALGEBRA-010 · ★★★ Advanced
How many distinct roots does \(z^4-2z^3+z^2\) have?
Choose one option.
Show answer
- B. \(2\)
Explanation
\(z^4-2z^3+z^2=z^2(z-1)^2\): the distinct roots are \(0\) and \(1\), both double; with multiplicity there are \(4\).
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 \(a_n\).
\[p(z)=a_n\prod_{k=1}^{n}(z-z_k)\]
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