20.5. Irreducible Polynomials

Try these first

1. 1.

   What is the definition of an irreducible polynomial?

   Hint

   Think about polynomials contributing to factorization.

2. 2.

   Identify whether (x² + 3) is irreducible in real numbers.

   Hint

   Consider possible factor pairs.

3. 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

   Hint

   Focus on the definition of irreducibility.

4. 4.

   True or False: All polynomials are reducible.

   • True
   • False

   Hint

   Recall examples of irreducible polynomials.

5. 5.

   Determine the irreducibility of the polynomial p(x) = x³ + x² + 1 in Z₂ and justify your answer.

   Hint

   Test it against available values in Z₂.

6. 6.

   Given q(x) = x⁴ + 2x² + 1, prove it is reducible in Z₄.

   Hint

   Factor it step-by-step.