23.2.2.2. Conditions for Quadratic Factors

Try these first

1. 1.

   What is the definition of a root of a polynomial?

   Hint

   Think about the values that make the polynomial equal zero.

2. 2.

   How many roots can a polynomial of degree 3 have?

   Hint

   The number of roots corresponds to the degree of the polynomial.

3. 3.

   What is the maximum number of roots a polynomial of degree 5 can have?

   • 5
   • 4
   • 6

   Hint

   Remember the definition of polynomial degree.

4. 4.

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

   • True
   • False

   Hint

   Think about the properties of monic polynomials and factorization.

5. 5.

   Show that the polynomial 2x^3 + 4x^2 + 2x can be factored into irreducible factors.

   Hint

   Look for common factors first.

6. 6.

   Prove that a polynomial of degree n will have at least one root in a field.

   Hint

   Consider how polynomials behave on an interval.