3 6 Fundamental Theorem of Algebra Objectives Use

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. Holt Mc. Dougal 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 Mc. Dougal Algebra 2

3 -6 Fundamental Theorem of Algebra Check It Out! Example 1 a Write the simplest polynomial function with the given zeros. – 2, 2, 4 Holt Mc. Dougal Algebra 2

3 -6 Fundamental Theorem of Algebra Check It Out! Example 1 b Write the simplest polynomial function with the given zeros. 0, 2 3 , 3 Holt Mc. Dougal 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 Mc. Dougal 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 Mc. Dougal Algebra 2

3 -6 Fundamental Theorem of Algebra Check It Out! Example 2 Solve x 4 + 4 x 3 – x 2 +16 x – 20 = 0 by finding all roots. Holt Mc. Dougal Algebra 2

3 -6 Fundamental Theorem of Algebra Check It Out! Example 2 Continued Step 2 Graph y = x 4 + 4 x 3 – x 2 + 16 x – 20 to find the real roots. Find the real roots at or near – 5 and 1. Holt Mc. Dougal Algebra 2

3 -6 Fundamental Theorem of Algebra Check It Out! Example 2 Continued Step 3 Test the possible real roots. Holt Mc. Dougal Algebra 2

3 -6 Fundamental Theorem of Algebra Check It Out! Example 3 Write the simplest function with zeros 2 i, 1+ and 3. Step 1 Identify all roots. Holt Mc. Dougal Algebra 2 2,

3 -6 Fundamental Theorem of Algebra Check It Out! Example 4 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. 1 Understand the Problem The cylinder and the hemisphere will have the same radius x. The answer will be the value of x. 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. Holt Mc. Dougal Algebra 2

3 -6 Fundamental Theorem of Algebra 2 Make a Plan Write an equation to represent the volume of the body of the silo. V = Vhemisphere + Vcylinder Holt Mc. Dougal Algebra 2

3 -6 Fundamental Theorem of Algebra 3 Solve Holt Mc. Dougal Algebra 2