1. Conjugate Pairs Let f (x) is a polynomial function that has real
coefficients. If a + bi, where b ≠ 0, is a zero
of the function, the conjugate a – bi is also
a zero of the function.
Ex 1: Find a fourth degree polynomial with real
coefficients that has -1, -1, and 3i as zeros.

2. Prime/Irreducible Factors
Every polynomial of degree n > 0 with real
coefficients can be written as the product of
linear and quadratic factors with real
coefficients, where the quadratic factors have
no real zeros.
A quadratic factor with no real zeros is said to be
prime or irreducible over the reals.

3. Ex 2: Write the polynomial f (x) = x4 – x2 – 20
a. As the product of factors irreducible over
the rationals.
b. As a product of linear factors.
c. As the product of linear factors and
quadratic factors that are irreducible over
the reals.

4. Ex 3: Find all the zeros of
f (x) = x4 – 3x3 + 6x2 + 2x – 60
given that 1 + 3i is a zero of f .

5. Practice Assignment:
Section 2.5B (pg 141) #35 – 52