All Zeros of Polynomial Functions

When you are done with this section, you will be able to do the following

Use the Fundamental Theorem of Algebra and the Linear Factorization Theorem to write a polynomial as the product of linear factors.
Find all real and complex zeros of a polynomial function.
Find a polynomial with integer coefficients whose zeros are given.
Use the Leading Coefficient Test and the zeros of a polynomial to sketch the graph of a polynomial.
Apply techniques for approximating real zeros to solve an application problem.

We are now putting together everything from Quadratic Polynomial Functions through Complex Numbers. We are going to find all the zeros (real and complex) of polynomial functions. Once we have all the zeros, we can write the polynomial as a product of linear factors.

Right now the best place to find information about complex numbers is your text book (pages 346-354 in College Algebra 3rd edition by Stewart, Redlin, and Watson).

Read carefully:

Fundamental Theorem of Algebra

Complete Factorization Theorem

Zeros Theorem

Conjugate Zeros Theorem

Additional Fundamental Theorem of Algebra Resources On-line Resources:

Complex Numbers/Fundamental Theorem of Algebra (starting on page 6)

Spark Notes: Complex Zeros and the Fundamental Theorem of Algebra

Complex Coefficients and Zeros

Zeros of Polynomial Functions

Coolmath Algebra

Reviewing the Quadratic Formula On-line Notes

Purplemath - Your Algebra Rexource: The Quadratic Formula Lesson - I

All Zeros of Polynomial Functions Homework:

The All Zeros of Polynomial Functions Homework contains 5 problems.
Things to remember:

You should start looking for roots either using your calculator or the Rational Root Test and synthetic division.

In a polynomial with integer coefficient, complex zeros occur in conjugate pairs.

The Fundamental Theorem of Algebra says that there are the same number of roots (zeros) as the degree of the polynomial. It does not say that the roots are all different. Repeated roots count as many times as their multiplicity.

Homework examples:

Example 1: similar to problem 2

Example 2: similar to problem 2

Example 3: similar to problem 3

Example 4: similar to problem 3

Example 5: simliar to problem 5

Keep in mind that it is recommended that you also complete all the problems in the set called "also recommended" with the same number as the All Zeros of Polynomial Functions Homework. There are many different ways to ask the same questions. This will allow you to see additional problems that are related to this topic.