Exercise · ZW-FUNDAMENTAL-THEOREM-OF-ALGEBRA-004 · ★☆☆ Basic
If \(p(z)=z^3+z+1\) has one real root, how many non-real roots does it have?
Choose one option.
Show answer
- B. \(2\)
Explanation
There are \(3\) roots in total; since the coefficients are real, non-real roots come in conjugate pairs: \(2\) remain.
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-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-009 In the proof via Liouville, if p had no roots, which function would be entire and bounded?
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