1. Section 3-6 Fundamental Theorem of Algebra
Objectives
Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots
Identify all of the roots of a polynomial equation

2. Fundamental Theorem of Algebra
You have learned several important properties about real roots of polynomial equations
You can use this information to write polynomial function when given in zeros
Create a binomial (x – zero)

3. Writing Polynomial Functions
Write the simplest polynomial with roots –1, , and 4
Write the simplest polynomial function with roots -2, 2, and 4 2 3

4. Fundamental Theorem of Algebra
Notice that the degree of the function in Example 1 is the same as the number of zeros. This is true for all polynomial functions. However, all of the zeros are not necessarily real zeros. Polynomial functions, like quadratic functions, may have complex zeros that are not real numbers.

5. Fundamental Theorem of Algebra
Using this theorem, you can write any polynomial function in factored form
To find all roots of a polynomial equation, you can use a combination of the Rational Root Theorem, the Irrational Root Theorem, and methods for finding complex roots, such as the quadratic formula
List possible zeros
Use calculator for 1 zero
Use synthetic with that zero
Factor the remaining polynomial
Set each factor = zero and solve

6. Finding All Roots of a Polynomial
Solve x4 – 3x3 + 5x2 – 27x – 36 = 0 by finding all roots
Solve x4 + 4x3 – x2 +16x – 20 = 0 by finding all roots

7. Complex Conjugate Root Theorem
If 𝑎+𝑏𝑖 is a root of a polynomial equation, then 𝑎−𝑏𝑖 is also a root
You may only be given one complex number as a root and you HAVE to include its conjugate as a root

8. Writing a Polynomial Function with Complex Zeros
Write the simplest function with zeros 2 + i, , and 1.

9. Problem Solving Application
A silo is in the shape of a cylinder with a cone-shaped top. The cylinder is 20 feet tall. The height of the cone is 1.5 times the radius. The volume of the silo is 828 cubic feet. Find the radius of the silo.
The cylinder and the cone have the same radius x. The answer will be the value of x

10. HOMEWORK
Page #12-52even, 57-62
Accelerated: Add #63-69