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Chapter 3
3-6 Fundamental Algebra theorem
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.
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.
Example #1
 Write the simplest polynomial with roots –1, 2 ,
3
and 4.
Example # 2
 Write the simplest polynomial function with the
given zeros.
–2, 2, 4
Example#3
 Write the simplest polynomial function with the
given zeros.
0,
2
3
,3
Student guided practice
 Do problems 1-3 page 193
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.
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.
Example
 Solve x4 – 3x3 + 5x2 – 27x – 36 = 0 by finding all
roots.
Student guided practice
 Do problem 4-6 page 193
Example writing equations
 Write the simplest function with zeros 2 + i,
and 1.
,
Example writing equation
 Write the simplest function with zeros 2i,
and 3.
,
Example 4: Problem-Solving Application
 A silo is in the shape of a cylinder with a coneshaped 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.
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.
Homework
 Do problems 11-20 in your book page 193
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