20.6. Factor Theorem for Polynomials Over Fields

Try these first

1. 1.

   Evaluate f(3) for f(x) = x^2 - 9 before identifying a factor.

   Hint

   Plug in x = 3 into the polynomial.

2. 2.

   Check if (x - 1) is a factor of f(x) = x^2 - x.

   Hint

   Evaluate f(x) at x = 1.

3. 3.

   What does the Factor Theorem state about a polynomial and its value at a root?

   • The polynomial is always positive
   • It has a factor
   • It cannot be evaluated

   Hint

   Remember what happens at roots.

4. 4.

   If f(0) = 5, does (x - 0) divide the polynomial f?

   • True
   • False

   Hint

   Consider how factors relate to roots.

5. 5.

   Prove that if a polynomial of degree n has n distinct roots, then it can be expressed as a product of linear factors.

   Hint

   Break down the proof by induction over the degree.

6. 6.

   Given a polynomial p(x) = x^4 - 16, find its factors and confirm the roots using the Factor Theorem.

   Hint

   Consider how to approach factorization in steps.