20.1. Division of Polynomials Over Fields

Try these first

1. 1.

   Provide an example of a polynomial and find its division with another polynomial.

   Hint

   Use polynomial long division.

2. 2.

   What is the degree of the polynomial 4x^3 - 2x + 7?

   Hint

   Look for the highest exponent.

3. 3.

   What is polynomial division?

   • Finding area under a curve
   • Dividing one polynomial by another to find quotient and remainder
   • Adding two polynomials

   Hint

   Think about how division works in arithmetic.

4. 4.

   True or False: All polynomials can be factored into irreducible polynomials.

   • True
   • False

   Hint

   Consider the factorization theorem.

5. 5.

   Find the GCD of the polynomials 3x^2 - 3x + 1 and 2x^2 + x - 1. Show your work.

   Hint

   Remember to apply the division step repeatedly.

6. 6.

   Prove that x^2 + 2x + 1 is reducible and express it in factored form.

   Hint

   Use the method of completing the square or recognize it as a perfect square trinomial.