1. Factoring with Real Number Coefficients
Sec. 2.6c
Homework: p odd, odd

2. Factors of a Polynomial with Real Coefficients
Definition: A quadratic with no real zeros
is irreducible over the reals.
The New Rule:
Every polynomial function with real coefficients
can be written as a product of linear factors
and irreducible quadratic factors, each with
real coefficients.

3. A Practice Problem Write the given function as a product of linear and
irreducible quadratic factors, each with real coefficients.
Graph to find a rational zero.
 x = 2/3
2/3 3 –2 6 –4
–24 16 2 4
–16 3 6
–24

4. A Practice Problem Write the given function as a product of linear and
irreducible quadratic factors, each with real coefficients.
How do we factor
this last term?

5. The zeros of the last term are complex…
A Practice Problem
Write the given function as a product of linear and
irreducible quadratic factors, each with real coefficients.
The zeros of the last term are complex…
We have taken the factorization of f as far as
possible, subject to the condition that each
factor has real coefficients.

6. Polynomial Function of Odd Degree
A function’s complex zeros always occur in
conjugate pairs…
Since a function with odd degree has an odd
number of zeros, what can we conclude?...
Every polynomial function of odd
degree with real coefficients has at
least one real zero.
Does this hold with our previous example???

7. More Guided Practice
Write the given function as a product of linear and
irreducible quadratic factors, each with real coefficients.

8. Whiteboard Practice
Write the given function as a product of linear and
irreducible quadratic factors, each with real coefficients.

9. Whiteboard Practice
Write the given function as a product of linear and
irreducible quadratic factors, each with real coefficients.