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17.4: Reading Questions

Last updated

Sep 29, 2021
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- 17.3: Irreducible Polynomials
- 17.5: Exercises

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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.

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