23.1.2. Number of Roots for Degree n Polynomial

Try these first

1. 1.

   What is a root of a polynomial?

   Hint

   Remember the definition of a root.

2. 2.

   What is the degree of the polynomial f(x) = 3x⁵ + 2x⁴ - x + 7?

   Hint

   Look for the term with the highest power.

3. 3.

   How many roots can a polynomial of degree n have?

   • n-1
   • n
   • n+1

   Hint

   Consider the definition of degree.

4. 4.

   Is the statement 'A quadratic polynomial can have at most two linear factors' true or false?

   • True
   • False

   Hint

   Think about the degree again.

5. 5.

   Consider the polynomial f(x) = x⁵ - 5x³ + 4x. Factor it completely over the reals and identify its roots.

   Hint

   Look for common factors first.

6. 6.

   Demonstrate that the polynomial f(x) = 2x⁴ + x³ - 4x² - 2x + 1 is irreducible over the rationals.

   Hint

   Try picking potential rational roots to start.