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Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
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.
Objectives
Holt McDougal Algebra 2
3-6 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.
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Write the simplest polynomial function with the given zeros.
Check It Out! Example 1a
–2, 2, 4
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Write the simplest polynomial function with the given zeros.
Check It Out! Example 1b
0, , 32 3
Holt McDougal Algebra 2
3-6 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. Polynomials functions, like quadratic functions, may have complex zeros that are not real numbers.
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
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.
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Solve x4 + 4x3 – x2 +16x – 20 = 0 by finding all roots.
Check It Out! Example 2
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Find the real roots at or near –5 and 1.
Step 2 Graph y = x4 + 4x3 – x2 + 16x – 20 to find the real roots.
Check It Out! Example 2 Continued
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Step 3 Test the possible real roots.
Check It Out! Example 2 Continued
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Write the simplest function with zeros 2i, , and 3.
Step 1 Identify all roots.
Check It Out! Example 3
1+ 2
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
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.
List the important information:
• The cylinder is 20 feet tall.
• The height of the hemisphere is x.
• The volume of the silo is 2106 cubic feet.
11 Understand the ProblemThe cylinder and the hemisphere will have the same radius x. The answer will be the value of x.
Check It Out! Example 4
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
22 Make a Plan
Write an equation to represent the volume of the body of the silo.
V = Vhemisphere + Vcylinder
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Solve33
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