23.2. Finding Irreducible Factors

Try these first

1. 1.

   Identify the root of f(x) = x - 3.

   Hint

   What value makes f(x) equal to 0?

2. 2.

   What defines a monic polynomial?

   Hint

   Consider the highest degree term.

3. 3.

   A polynomial of degree n can have at most how many roots?

   • n-1
   • n
   • n+1

   Hint

   Consider how roots relate to the degree.

4. 4.

   True or False: Every polynomial has at least one real root.

   • True
   • False

   Hint

   Think about polynomials with even degrees and positive leading coefficients.

5. 5.

   Factor the polynomial f(x) = x^4 - 5x^2 + 6.

   Hint

   Try substituting a new variable to simplify the expression.

6. 6.

   Prove that the polynomial x^5 + x + 1 has no rational roots.

   Hint

   Evaluate potential candidates and see if f(α) = 0 is ever true for them.