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Polynomial Equations and Finding Roots: Objectives, Examples, and Techniques

This article covers the objectives of using the Fundamental Theorem of Algebra and its corollary to write polynomial equations of least degree with given roots, as well as identifying all the roots of a polynomial equation. It also provides relevant examples and techniques for finding real and complex roots of polynomial functions.

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Polynomial Equations and Finding Roots: Objectives, Examples, and Techniques

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  1. 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.

  2. 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.

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

  4. 2 3 Check It Out! Example 1b Write the simplest polynomial function with the given zeros. 0, , 3

  5. 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.

  6. 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.

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

  8. 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.

  9. Check It Out! Example 2 Continued Step 3 Test the possible real roots.

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

  11. 1 Understand the Problem 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. 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.

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

  13. 3 Solve

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