19.2.9. Polynomials Over Rings

Try these first

1. 1.

   Define a polynomial of degree 2 over a ring.

   Hint

   Consider the highest power term.

2. 2.

   What does closure property mean in the context of polynomials over a ring?

   Hint

   Think about the operations performed.

3. 3.

   What is the degree of the polynomial P(x) = 5x^3 + 4x + 2?

   • 1
   • 2
   • 3

   Hint

   Look at the power of x terms.

4. 4.

   True or False: The sum of two polynomials of degrees 2 and 3 will always yield a polynomial of degree 3.

   • True
   • False

   Hint

   Think about which degree dominates.

5. 5.

   Create two polynomials over a ring defined by integers modulo 4 and show their addition and multiplication. What is the result of each operation?

   Hint

   Keep track of modulo operations throughout.

6. 6.

   Prove that the product of a polynomial of degree m and another of degree n has a degree of m+n.

   Hint

   Visualize the polynomial terms’ expansion.