23.1. Roots of a Polynomial

Try these first

1. 1.

   What is the definition of a root of a polynomial?

   Hint

   Think about what happens to the polynomial when you substitute this value.

2. 2.

   How many roots can a polynomial of degree 3 have?

   Hint

   Consider the maximum number of linear factors it can have.

3. 3.

   What defines a root of a polynomial?

   Hint

   Think about the output of the polynomial when you substitute the value.

4. 4.

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

   • True
   • False

   Hint

   Remember the relationship between degree and roots.

5. 5.

   Prove that a quadratic polynomial f(x) = ax^2 + bx + c can have zero, one, or two roots based on its discriminant.

   Hint

   Analyze the discriminant carefully!

6. 6.

   Given f(x) = x^4 + 1, demonstrate its factorization into irreducible quadratics over the field of real numbers and integers.

   Hint

   Keep track of coefficients as you set up your equations!