### 1.

Suppose $$p(x)$$ is a polynomial of degree $$n$$ with coefficients from any field. How many roots can $$p(x)$$ have? How does this generalize your high school algebra experience?

### 2.

What is the definition of an irreducible polynomial?

### 3.

Find the remainder upon division of $$8 \, x^{5} - 18 \, x^{4} + 20 \, x^{3} - 25 \, x^{2} + 20$$ by $$4 \, x^{2} - x - 2\text{.}$$

### 4.

A single theorem in this chapter connects many of the ideas of this chapter to many of the ideas of the previous chapter. State a paraphrased version of this theorem.

### 5.

Early in this chapter, we say, “We can prove many results for polynomial rings that are similar to the theorems we proved for the integers.” Write a short essay (or a very long paragraph) justifying this assertion.