Practice Irreducible Polynomials (20.5) - Polynomials Over Fields and Properties

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9 questions

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Exercise
6 questions · about 4 min · pass mark 70%. Hints available.
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6 sample questions

Question 1.
What is the definition of an irreducible polynomial?
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Think about polynomials contributing to factorization.
Question 2.
Identify whether (x² + 3) is irreducible in real numbers.
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Consider possible factor pairs.
3Question 3.
What is an irreducible polynomial?
- A polynomial that can be factored
- A non-constant polynomial with no non-constant factors
- A polynomial that can be divided by constant only
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Focus on the definition of irreducibility.
4Question 4.
True or False: All polynomials are reducible.
- True
- False
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Recall examples of irreducible polynomials.
Question 5.
Determine the irreducibility of the polynomial p(x) = x³ + x² + 1 in Z₂ and justify your answer.
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Test it against available values in Z₂.
Question 6.
Given q(x) = x⁴ + 2x² + 1, prove it is reducible in Z₄.
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Factor it step-by-step.

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