Exercise · ZW-FUNDAMENTAL-THEOREM-OF-ALGEBRA-009 · ★★★ Advanced
In the proof via Liouville, if \(p\) had no roots, which function would be entire and bounded?
Choose one option.
Show answer
- C. \(1/p\)
Explanation
\(1/p\) would be entire (no zeros to produce poles) and \(|1/p(z)|\to0\) as \(|z|\to\infty\), hence bounded; by Liouville it would be constant, and so would \(p\): a contradiction.
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
- ZW-FUNDAMENTAL-THEOREM-OF-ALGEBRA-001 How many complex roots, counted with multiplicity, does z^5-3z+1 have?
- ZW-FUNDAMENTAL-THEOREM-OF-ALGEBRA-002 The roots of z^2+4 are:
- ZW-FUNDAMENTAL-THEOREM-OF-ALGEBRA-003 What are the four roots of z^4-1?
- ZW-FUNDAMENTAL-THEOREM-OF-ALGEBRA-004 If p(z)=z^3+z+1 has one real root, how many non-real roots does it have?
- ZW-FUNDAMENTAL-THEOREM-OF-ALGEBRA-005 The roots of z^3-6z^2+11z-6 add up to:
- ZW-FUNDAMENTAL-THEOREM-OF-ALGEBRA-006 The product of the three roots of z^3+2z^2-z-10 is:
- 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-008 ZW-FUNDAMENTAL-THEOREM-OF-ALGEBRA-010 →