23.2.2. Example of a Polynomial Factorization

Try these first

1. 1.

   Define what a root of a polynomial is.

   Hint

   Think of simple polynomials and what values make them zero.

2. 2.

   What does it mean for a polynomial to be monic?

   Hint

   Look at the term with the highest degree.

3. 3.

   What is a root of a polynomial?

   Hint

   Consider simple examples of roots.

4. 4.

   True or False: A degree n polynomial can have more than n roots.

   • True
   • False

   Hint

   Think of the definition of polynomial degree.

5. 5.

   Prove algebraically that a polynomial of degree 5 can have at most 5 roots.

   Hint

   Use polynomial long division.

6. 6.

   Given the polynomial x⁵ + x + 1, determine its irreducibility over the integers.

   Hint

   Test possible rational roots derived from factors of the constant term.