234 CHAPTER 4 Polynomial and Rational Functions Repeat Step 4 The depressed equation − + − = x x x 3 27 9 0 3 2 can be factored by grouping. ( ) ( ) ( ) ( ) − + − = − + − = + − = + = − = = − = = − = = x x x x x x x x x x x x x i x i x 3 27 9 0 3 1 9 3 1 0 9 3 1 0 9 0 or 3 1 0 9 or 3 1 3 , 3 or 1 3 3 2 2 2 2 2 The four complex zeros of f are − − i i 3, 3, 2, and 1 3 . The factored form of f is f x x x x x x i x i x x x i x i x x 3 5 25 45 18 3 3 2 3 1 3 3 3 2 1 3 4 3 2 ( ) ( ) ( )( )( )( ) ( )( )( ) = + + + − = + − + − = + − + − Now Work PROBLEM 35 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 4.4 Assess Your Understanding 1. Find the sum and the product of the complex numbers i 3 2 − and i 3 5 . − + (pp. A59–A63) 2. Solve x x2 2 0 2 + + = in the complex number system. (pp. A63–A65) 3. What is the conjugate of i 3 4 ? − + (p. A60) 4. Given z i 5 2 , = + find the product z z. ⋅ (p. A60) Skill Building In Problems 9–18, information is given about a polynomial function f whose coefficients are real numbers. Find the remaining zeros of f . 9. Degree 3; zeros: i 3, 4 − 10. Degree 3; zeros: i 4, 3 + 11. Degree 4; zeros: i i , 3 + 12. Degree 4; zeros: 1, 2, i 2 + 13. Degree 5; zeros: 1, i , i5 14. Degree 5; zeros: 0, 1, 2, i 15. Degree 4; zeros: i, 7, 7− 16. Degree 4; zeros: i i 2 , − − 17. Degree 6; zeros: i i 2, 4 9, 7 2, 0 + − − 18. Degree 6; zeros: i i i , 3 2 , 2 − − + 5. Every polynomial function of odd degree with real coefficients will have at least real zero(s). 6. If i 3 4 + is a zero of a polynomial function of degree 5 with real coefficients, then so is . 7. True or False A polynomial function of degree n with real coefficients has exactly n complex zeros. At most n of them are real zeros. 8. True or False A polynomial function of degree 4 with real coefficients could have i i 3, 2 , 2 , − + − and i 3 5 − + as its zeros. Concepts and Vocabulary In Problems 19–24, find a polynomial function f with real coefficients having the given degree and zeros. Answers will vary depending on the choice of leading coefficient. 19. Degree 4; zeros: i 3 2 ; + 4, multiplicity 2 20. Degree 4; zeros: i i , 1 2 + 21. Degree 5; zeros: 2; i i ; 1 − + 22. Degree 6; zeros: i i i , 4 ; 2 − + 23. Degree 4; zeros: 3, multiplicity 2; i− 24. Degree 5; zeros: 1, multiplicity 3; i 1 + 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure Factor x2 from − x x 3 3 2 and 9 from −x 27 9. Factor out the common factor −x3 1. Use the Zero-Product Property.
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