Math 361, Spring 2015, Assignment 2

Carefully define the following terms, then give one example and one non-example of each:

1. Function induced by a polynomial.
2. Root of a polynomial.
3. Quotient and remainder (when one polynomial is divided by another).
4. Divisor (of a polynomial).
5. Irreducible polynomial.

Carefully state the following theorems (you do not need to prove them):

1. Theorem on polynomial long division (Theorem 23.1)
2. Factor theorem.
3. Theorem concerning the number of roots of a polynomial (Corollary 23.5).
4. Theorem concerning finite subgroups of the group of units of a field (Theorem 23.6).
5. Theorem concerning irreducibility of quadratic and cubic polynomials (Theorem 23.10).
6. Theorem on unique factorization in $F[x]$.
7. Rational root theorem (Corollary 23.12 is a special case; we gave the general version in class).
8. Eisenstein's Criterion.

Solve the following problems:

1. Section 23, problems 1, 5, 9, 13, 19, and 21.

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