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Fundamental theorem of algebra - Quiz
Contributed by: Jordan
  • 1. What kind of polynomial must have at least one complex root according to the Fundamental Theorem of Algebra?
A) Linear polynomial of zero degree
B) Univariate polynomial of positive degree
C) Bivariate polynomial of negative degree
D) Constant polynomial of positive degree
  • 2. If a polynomial has real coefficients, what can be said about its roots?
A) It cannot have complex roots
B) It has at least one complex root
C) It has no roots
D) It must have only real roots
  • 3. What is the result of multiplying a polynomial with complex coefficients by its complex conjugate?
A) A polynomial with zero degree
B) A polynomial with real coefficients
C) A polynomial with no roots
D) A polynomial with imaginary coefficients
  • 4. How can a univariate polynomial of degree n with complex coefficients be factorized?
A) c(x+r1)⋯(x+rn)
B) c(x−r1)n
C) c(x−r1)⋯(x−rn)
D) c(xn - r1)
  • 5. What term describes the number of occurrences of a root in a factored polynomial?
A) Degree
B) Coefficient
C) Multiplicity
D) Constant
  • 6. How is the existence of multiple roots proved from the general theorem?
A) Quadratic division
B) Subtraction of constants
C) Induction on real numbers
D) Recursion on n
  • 7. According to the complex conjugate root theorem, if r is a non-real root of a polynomial with real coefficients, what else is a root?
A) The reciprocal of r
B) The negative of r
C) The square of r
D) The complex conjugate of r
  • 8. What is the degree of the polynomial formed by multiplying (x−r) and (x−r‾) where r is a non-real root?
A) One
B) Two
C) Four
D) Three
  • 9. Every univariate polynomial with real coefficients of degree larger than two has what?
A) No real roots
B) A factor of degree one with complex coefficients
C) A degree three factor
D) A factor of degree two with real coefficients
  • 10. What tools are typically required for proofs of the Fundamental Theorem of Algebra?
A) Strictly algebraic manipulation
B) Mathematical analysis or topological continuity
C) Basic arithmetic only
D) Only Euclidean geometry
  • 11. Why is it sometimes claimed the Fundamental Theorem of Algebra is not a theorem of algebra?
A) Because proofs involve analysis or continuity
B) Because algebra is not used at all
C) Because it has no proof
D) Because it only applies to real numbers
  • 12. In the context of the 'growth lemma', how does a polynomial p(z) of degree n behave when |z| is very large?
A) It behaves like a constant
B) It behaves like 1/z
C) It behaves like zn
D) It behaves like zero
  • 13. What is the relationship between real numbers and complex numbers in the context of the theorem?
A) Real numbers are also complex numbers
B) Complex numbers are a subset of real numbers
C) They are mutually exclusive sets
D) They have no relationship
  • 14. If p is a polynomial with complex coefficients, what is p‾?
A) The derivative of p
B) The integral of p
C) The reciprocal of p
D) A polynomial with conjugated coefficients
  • 15. What is the purpose of the monic polynomials pi in the factorization of real polynomials?
A) Each is of degree at most two
B) Each must be a prime number
C) Each must be of degree n
D) Each must be of degree zero
  • 16. What is another name for the Fundamental Theorem of Algebra?
A) Euclid's theorem
B) Riemann's theorem
C) Newton's theorem
D) d'Alembert's theorem
  • 17. Which mathematician wrote in 1608 that a polynomial of degree n may have n solutions?
A) Albert Girard
B) Carl Friedrich Gauss
C) Leonhard Euler
D) Peter Roth
  • 18. Who famously made an erroneous claim in 1702 regarding the factorization of x4 + a4?
A) Euler
B) Lagrange
C) Leibniz
D) Bernoulli
  • 19. Whose proof of the fundamental theorem of algebra, published in 1806, was the first rigorous one?
A) Cauchy
B) Argand
C) Gauss
D) Weierstrass
  • 20. Which mathematician first raised the problem of finding a constructive proof for the theorem?
A) Argand
B) Kneser
C) Weierstrass
D) Gauss
  • 21. The fundamental theorem of algebra cannot be proven using purely algebraic methods because it requires which of the following?
A) Fundamental theorem of calculus
B) Pythagorean theorem
C) Intermediate value theorem
D) Binomial theorem
  • 22. Who filled the topological gap in Gauss's 1799 proof in the year 1920?
A) Hellmuth Kneser
B) Martin Kneser
C) James Wood
D) Alexander Ostrowski
  • 23. In which textbook was the first proof of the theorem included?
A) Principia Mathematica
B) Cours d'analyse
C) L'invention nouvelle en l'Algèbre
D) Arithmetica Philosophica
  • 24. What type of functions are Rp(x)(a, b) and Sp(x)(a, b) in terms of a and b?
A) Trigonometric functions
B) Constant values
C) Bivariate polynomials
D) Linear functions
  • 25. Who authored the textbook 'Linear Algebra Done Right' which contains a proof of the fundamental theorem of algebra?
A) Gauss
B) Sheldon Axler
C) Sturm
D) De Moivre
  • 26. Which mathematical tool is mentioned in relation to the interlacing property of the chain?
A) A Sturm-like chain
B) Bayes' theorem
C) The Pythagorean theorem
D) The fundamental theorem of calculus
  • 27. In Sheldon Axler's proof, which formula is used?
A) De Moivre's formula
B) Quadratic formula
C) Binomial theorem
D) Euler's identity
  • 28. What property of the roots of Rp(x)(a, b) and Sp(x)(a, b) in the variable a is used for large negative values of b?
A) Complex conjugation
B) Irrationality
C) Interlacing property
D) Parity property
  • 29. According to the maximum modulus principle, what can be concluded about 1/p(z) at z0?
A) 1/p(z) has no zeros
B) p(z0) is a local maximum
C) p(z0) = 0
D) p(z0) = 1
  • 30. In the proof not requiring the maximum modulus principle, what does q(z) represent?
A) p(z) - a
B) ck(z - z0)n
C) a + ck(z - z0)k
D) p(z)/a
  • 31. Liouville's theorem states that a bounded entire function must be:
A) Constant
B) Zero
C) Unbounded
D) Differentiable
  • 32. The resolvent function R(z) is defined as:
A) (A - zIn)-1
B) (zA - In)-1
C) (zIn - A)-1
D) (In - zA)-1
  • 33. In the induction proof, when k=0, what is the degree n of the polynomial?
A) Zero.
B) Odd.
C) Even.
D) Prime.
  • 34. Who showed in 2007 that the assumption that odd degree polynomials have roots is stronger than necessary?
A) Galois.
B) The text does not name the person.
C) Sylow.
D) Joseph Shipman.
  • 35. Who developed the geometric proof for the fundamental theorem of algebra mentioned in the text?
A) Jean d'Alembert
B) Carl Friedrich Gauss
C) J. M. Almira and A. Romero
D) Augustin-Louis Cauchy
  • 36. What geometric concept is used to prove the theorem in this text?
A) Hyperbolic topology
B) Fractal geometry
C) Riemannian geometric arguments
D) Euclidean triangle geometry
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