20. Polynomials Over Fields and Properties

Try these first

1. 1.

   What is the remainder when dividing 2x^3 + x by x + 1?

   Hint

   Use synthetic division method.

2. 2.

   True or False: All polynomials are reducible.

   Hint

   Think of irreducible polynomials.

3. 3.

   What is the result when you divide a polynomial by itself?

   • Remainder is zero
   • The original polynomial
   • Quotient is zero

   Hint

   Remember division properties.

4. 4.

   True or False: The polynomial x^2 + 1 is irreducible over the real numbers.

   • True
   • False

   Hint

   Think about solving for roots.

5. 5.

   Given the polynomial f(x) = x^3 - 7x + 6, determine if it’s irreducible over the rationals.

   Hint

   Evaluate potential rational roots.

6. 6.

   Find the GCD of f(x) = x^4 + 2 and g(x) = x^2 + 1 using the Euclidean algorithm.

   Hint

   Perform polynomial long division.