23.1.2.1. Proof of Roots Upper Bound

Try these first

1. 1.

   Define a root of a polynomial.

   Hint

   Think about the equation f(x) = 0.

2. 2.

   What does it mean if a polynomial is monic?

   Hint

   Consider the highest degree term.

3. 3.

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

   • n-1
   • n
   • n+1

   Hint

   Remember the definition of polynomial degree.

4. 4.

   The factor theorem states that if f(α) = 0, then...

   • True
   • False

   Hint

   Think about how roots relate to factors.

5. 5.

   Demonstrate that if f(x) = x^n, where n is a positive integer, can yield n roots, and identify these roots.

   Hint

   Consider how many times you hit zero from the polynomial.

6. 6.

   Given the polynomial p(x) = x^3 - 6x^2 + 9x, factor it completely.

   Hint

   Look for patterns in coefficients that connect to roots.