23.2.1. Methods for Finding Irreducible Factors

Try these first

1. 1.

   What is a root of a polynomial?

   Hint

   Think about what happens when you substitute a number into the polynomial.

2. 2.

   Define a monic polynomial.

   Hint

   What is special about the leading coefficient?

3. 3.

   What does the factor theorem state?

   • If α is a root
   • then (x - α) is a factor.
   • All roots are irrational.
   • Polynomials cannot have real coefficients.

   Hint

   Think about what happens if substituting α into the polynomial results in zero.

4. 4.

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

   • True
   • False

   Hint

   Relate to polynomial properties.

5. 5.

   Given the polynomial x^5 + x^3 + x^2, find its irreducible factors over the field of integers.

   Hint

   Look for integers that satisfy the equation.

6. 6.

   Using the polynomial x^4 + 2x^2 + 1, identify and justify its potential irreducibility.

   Hint

   Check if your polynomial resembles the form of squares.