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36 Fundamental Theorem of Algebra period 1.notebook October 30, 2014 Bellwork 10-30-14 Identify all the real roots of each equation. 1. 4x5 – 8x4 – 32x3 = 0 2. x3 –x2 + 9 = 9x 1 36 Fundamental Theorem of Algebra period 1.notebook October 30, 2014 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 36 Fundamental Theorem of Algebra period 1.notebook October 30, 2014 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 36 Fundamental Theorem of Algebra period 1.notebook October 30, 2014 Example 1: Writing Polynomial Functions Write the simplest polynomial with roots –1, 2 , and 4. 3 If r is a zero of P(x), then x – r is a factor of P(x). Multiply the first two binomials. Multiply the trinomial by the binomial. 4 36 Fundamental Theorem of Algebra period 1.notebook October 30, 2014 Check It Out! Example 1a Write the simplest polynomial function with the given zeros. –2, 2, 4 If r is a zero of P(x), then x – r is a factor of P(x). 5 36 Fundamental Theorem of Algebra period 1.notebook October 30, 2014 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 36 Fundamental Theorem of Algebra period 1.notebook October 30, 2014 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 36 Fundamental Theorem of Algebra period 1.notebook October 30, 2014 Example 2: Finding All Roots of a Polynomial Solve x4 – 3x3 + 5x2 – 27x – 36 = 0 by finding all roots. The polynomial is of degree 4, so there are exactly four roots for the equation. Step 1 Use the rational Root Theorem to identify rational roots. Step 2 Graph y = x4 – 3x3 + 5x2 – 27x – 36 to find the real roots. 8 36 Fundamental Theorem of Algebra period 1.notebook October 30, 2014 Example 2 Continued Step 3 Test the possible real roots. x4 – 3x3 + 5x2 – 27x – 36 –1 9 36 Fundamental Theorem of Algebra period 1.notebook October 30, 2014 Example 2 Continued The polynomial factors into (x + 1)(x3 – 4x2 + 9x – 36) = 0. Step 4 Solve x2 + 9 = 0 to find the remaining roots. The fully factored form of the equation is: The solutions are: 10 36 Fundamental Theorem of Algebra period 1.notebook October 30, 2014 Homework: Textbook Page 193, #'s 1-5 odd and 11-19 odd 11