1. Chapter 3 3-6 Fundamental Algebra theorem

2. 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.

3. Fundamental algebra theorem  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.

4. Example #1  Write the simplest polynomial with roots –1,, and 4. 2 3

5. Example # 2  Write the simplest polynomial function with the given zeros. –2, 2, 4

6. Example#3  Write the simplest polynomial function with the given zeros. 0,, 3 2 3

7. Student guided practice  Do problems 1-3 page 193

8. Fundamental algebra theorem  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. Polynomials functions, like quadratic functions, may have complex zeros that are not real numbers.

9. Fundamental algebra theorem  Using this theorem, you can write any polynomial function in factor 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.

10. Example  Solve x 4 – 3x 3 + 5x 2 – 27x – 36 = 0 by finding all roots.

12. Student guided practice  Do problem 4-6 page 193

13. Example writing equations  Write the simplest function with zeros 2 + i,, and 1.

14. Example writing equation  Write the simplest function with zeros 2i,, and 3.

15. Example 4: 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.

16. Example  A grain silo is in the shape of a cylinder with a hemisphere top. The cylinder is 20 feet tall. The volume of the silo is 2106 cubic feet. Find the radius of the silo.

17. Homework  Do problems 11-20 in your book page 193

18. Closure  Today we learned about the fundament algebra theorem and how we find the zeros of functions.  Next class we are going to learned about graphs of polynomials