3.6 Alg II Objectives Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots. Identify.

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Presentation transcript:

3.6 Alg II 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.

3.6 Alg II 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.

Write the simplest polynomial function with the given zeros. 3.6 Alg II Check It Out! Example 1a Write the simplest polynomial function with the given zeros. –2, 2, 4

Write the simplest polynomial function with the given zeros. 3.6 Alg II Check It Out! Example 1b Write the simplest polynomial function with the given zeros. 0, , 3 2 3

3.6 Alg II 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.

3.6 Alg II 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.

Solve x4 + 4x3 – x2 +16x – 20 = 0 by finding all roots. 3.6 Alg II Check It Out! Example 2 Solve x4 + 4x3 – x2 +16x – 20 = 0 by finding all roots.

Check It Out! Example 2 Continued 3.6 Alg II Check It Out! Example 2 Continued Step 2 Graph y = x4 + 4x3 – x2 + 16x – 20 to find the real roots. Find the real roots at or near –5 and 1.

Check It Out! Example 2 Continued 3.6 Alg II Check It Out! Example 2 Continued Step 3 Test the possible real roots.

Write the simplest function with zeros 2i, , and 3. 1+ 2 3.6 Alg II Check It Out! Example 3 Write the simplest function with zeros 2i, , and 3. 1+ 2 Step 1 Identify all roots.

Understand the Problem 3.6 Alg II 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.

V = Vhemisphere + Vcylinder 3.6 Alg II 2 Make a Plan Write an equation to represent the volume of the body of the silo. V = Vhemisphere + Vcylinder

3.6 Alg II Solve 3