20.3. The GCD of Polynomials Over Fields

Try these first

1. 1.

   Define the GCD of two polynomials.

   Hint

   Think about the concept of GCD from integers.

2. 2.

   Give an example of an irreducible polynomial.

   Hint

   Consider polynomials you cannot factor over a given field.

3. 3.

   What is the GCD of two polynomials?

   • The smallest polynomial
   • The largest polynomial that divides both
   • Any polynomial that divides both

   Hint

   Think about what 'greatest' means.

4. 4.

   A polynomial is irreducible if:

   • True
   • False

   Hint

   Consider the definition of irreducibility.

5. 5.

   Find the GCD of x^4 - 1 and x^2 - 1 using polynomial division.

   Hint

   Start the division process carefully.

6. 6.

   Show that the polynomial 3x^3 + 6x^2 + 3x is reducible. Factor it completely.

   Hint

   Look for common factors first!