Math 360, Fall 2013, Assignment 15

I must study politics and war that my sons may have liberty to study mathematics and philosophy.

- John Adams, letter to Abigail Adams, May 12, 1780

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

1. Theorem concerning polynomial long division.
2. Factor theorem.
3. Theorem bounding the number of roots of a given polynomial.
4. Theorem on reducibility of quadratic and cubic polynomials.
5. Theorem relating factorization in $\mathbb{Q}[x]$ to factorization in $\mathbb{Z}[x]$.
6. Rational root theorem.
7. Eisenstein's criterion.

Solve the following problems:

1. Section 23, problems 1, 9, 15, 19, and 21.

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