23.2.2.3. Solving the Equations

Try these first

1. 1.

   What is a root of a polynomial?

   Hint

   Think of the definition we started with.

2. 2.

   What is the factor theorem?

   Hint

   Recall how roots relate to factors.

3. 3.

   What does the factor theorem state?

   • If f(α) = 0
   • then (x - α) is a factor of f(x).
   • Every polynomial has exactly n factors.
   • The degree of a polynomial does not affect the number of roots.

   Hint

   Think about how roots relate to the factors of polynomials.

4. 4.

   True or False: A polynomial of degree n can have n distinct roots.

   • True
   • False

   Hint

   Recall the relationship between degree and roots.

5. 5.

   Given the polynomial p(x) = 2x^3 - 4x^2 + 2x, find its roots and determine its irreducible components.

   Hint

   Consider using factoring techniques or the quadratic formula.

6. 6.

   Factor the polynomial q(x) = x^4 + x² + 1 over complex numbers.

   Hint

   Look for patterns or potential quadratic factors.