Algebra-Trigonometry-CynthiaYoung-3ed48.pdf - c04c.qxd 3:43 PM Page 441 4.5 Complex Zeros The Fundamental Theorem of Algebra Factoring a Polynomial

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4.5 Complex Zeros: The Fundamental Theorem of Algebra 441 S TEP 3 Utilize the rational zero theorem to determine the possible rational zeros. ; 1, ; 2 S TEP 4 Test possible rational zeros. 1 is a zero: 1 1 2 0 0 1 2 1 3 3 3 2 1 3 3 3 2 0 1 is a zero: 1 1 3 3 3 2 1 2 1 2 1 2 1 2 0 2 is a zero: 2 1 2 1 2 2 0 2 1 0 1 0 S TEP 5 Write P ( x ) as a product of linear factors. P ( x ) ( x 1)( x 1)( x 2)( x i )( x i ) = ( x - i )( x + i ) x 2 + 1 Technology Tip The graph of P ( x ) x 5 2 x 4 x 2 is shown. The real zeros of the function at x 2, x 1, and x 1 give the factors of x 2, x 1, and x 1. Use synthetic division to find the other factors. A table of values supports the real zeros of the function and its factors. Study Tip From Step 2 we know there is one positive real zero, so test the positive possible rational zeros first in Step 4. Study Tip P ( x ) is a fifth -degree polynomial, so we expect five zeros. SUMMARY SECTION 4.5 P ( x ) has at least one zero and no more than n zeros. If a bi is a zero, then a bi is also a zero. The polynomial can be written as a product of linear factors, not necessarily distinct. In this section we discussed complex zeros of polynomial functions. A polynomial function, P ( x ), of degree n with real coefficients has the following properties: t EXAMPLE 6 Factoring a Polynomial Factor the polynomial P ( x ) x 5 2 x 4 x 2. Solution: S TEP 1 Determine variations in sign. P ( x ) has 1 sign change. P ( x ) x 5 2 x 4 x 2 P ( x ) has 2 sign changes. P ( x ) x 5 2 x 4 x 2 S TEP 2 Apply Descartes’ rule of signs and summarize the results in a table. P OSITIVE N EGATIVE C OMPLEX R EAL Z EROS R EAL Z EROS Z EROS 1 2 2 1 0 4
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442 CHAPTER 4 Polynomial and Rational Functions In Exercises 1–8, find all zeros (real and complex). Factor the polynomial as a product of linear factors. 1. P ( x ) x 2 4 2. P ( x ) x 2 9 3. P ( x ) x 2 2 x 2 4. P ( x ) x 2 4 x 5 5. P ( x ) x 4 16 6. P ( x ) x 4 81 7. P ( x ) x 4 25 8. P ( x ) x 4 9 In Exercises 9–16, a polynomial function with real coefficients is described. Find all remaining zeros. 9. Degree: 3 Zeros: 1, i 10. Degree: 3 Zeros: 1, i 11. Degree: 4 Zeros: 2 i , 3 i 12. Degree: 4 Zeros: 3 i , 2 i 13. Degree: 6 Zeros: 2 (multiplicity 2), 1 3 i , 2 5 i 14. Degree: 6 Zeros: 2 (multiplicity 2), 1 5 i, 2 3 i 15. Degree: 6 Zeros: i , 1 i (multiplicity 2) 16. Degree: 6 Zeros: 2 i , 1 i (multiplicity 2) In Exercises 17–22, find a polynomial of minimum degree that has the given zeros. 17. 18. 19. 20. 21. 22. In Exercises 23–34, given a zero of the polynomial, determine all other zeros (real and complex) and write the polynomial in terms of a product of linear factors. Polynomial Zero Polynomial Zero 23. P ( x ) x 4 2 x 3 11 x 2 8 x 60 2 i 24. P ( x ) x 4 x 3 7 x 2 9 x 18 3 i 25. P ( x ) x 4 4 x 3 4 x 2 4 x 3 i 26. P ( x ) x 4 x 3 2 x 2 4 x 8 2 i 27. P ( x ) x 4 2 x 3 10 x 2 18 x 9 3 i 28. P ( x ) x 4 3 x 3 21 x 2 75 x 100 5 i 29. P ( x ) x 4 9 x 2 18 x 14 1 i 30. P ( x ) x 4 4 x 3 x 2 6 x 40 1 2 i 31. P ( x ) x 4 6 x 3 6 x 2 24 x 40 3 i 32. P ( x ) x 4 4 x 3 4 x 2 4 x 5 2 i 33. P ( x ) x 4 9 x 3 29 x 2 41 x 20 2 i 34. P ( x ) x 4 7 x 3 14 x 2 2 x 20 3 i In Exercises 35–58, factor each polynomial as a product of linear factors.
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