R_6 - Chapter R Review R.6 1. 3. 5. Polynomial Division;...

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Unformatted text preview: Chapter R Review R.6 1. 3. 5. Polynomial Division; Synthetic Division quotient, divisor, remainder True 2. 4. 4 x 2 -11x + 23 x+ 1 x 4 x 3 + 8x 2 -11x 2 + 2 -3) 2 0 - 5 True 1 x + 2 4 x 3 - 3x 2 + ) Check: (x + 2)(4x 2 - 11x + 23) + (- 45) = 4x 3 - 11x 2 + 23x + 8x 2 - 22x + 46 - 45 = 4x 3 - 3x 2 + x + 1 The quotient is 4x 2 - 11x + 23; the remainder is 45. -11x - 22x 23x + 1 23x + 46 - 45 6. 3 x 2 - 7x + 15 x + 2 3x 3 - x 2 + x - 2 ) 3x 3 + 6x 2 - 7x 2 + x - 7x 2 -14 x 15x - 2 15x + 30 - 32 Check: (x + 2)(3x 2 - 7x + 15) + (- 3 2 ) = 3x 3 - 7x 2 + 15x + 6x 2 - 14x + 30 - 32 = 3x 3 - x 2 + x - 2 The quotient is 3x 2 - 7x + 15; the remainder is 32. 7. x 2 4 x 3 - 3x 2 + x + 1 4x3 - 3x 2 -3x 2 x+ 1 ) 4x - 3 Check: (x 2 )(4 x - 3)+ (x + 1) = 4x 3 - 3x2 + x +1 The quotient is 4x - 3; the remainder is x +1. 30 Section R.6 8. 3x -1 3x 3 - x2 - x2 x- 2 Polynomial Division; Synthetic Division Check: (x 2 )(3x -1)+ (x - 2) = 3x 3 - x 2 + x - 2 The quotient is 3x -1 ; the remainder is x - 2. x 2 3x 3 - x 2 + x - 2 ) 9. x 2 + 2 5x 4 - 3x 2 + x + 1 5x 4 + 10x 2 -13x 2 + x + 1 -13x 2 - 26 x + 27 ) 5x 2 -13 Check : (x 2 + 2)(5x 2 -13) + (x + 27) = 5x 4 -13x 2 + 10x 2 - 26 + x + 27 = 5x 4 - 3x 2 + x + 1 The quotient is 5x 2 -13; the remainder is x + 27. Check : 2 10. 2 5x 2 -11 x + 2 5x - x + x - 2 4 ) (x 2 + 2)(5x 2 -11)+ (x + 20) = 5x 4 -11x 2 + 10x 2 - 22 + x + 20 = 5x 4 - x 2 + x - 2 The quotient is 5x 2 -11; the remainder is x + 20. Check : (2x 3 -1)(2x 2 ) + (-x 2 + x + 1) = 4 x 5 - 2x 2 - x 2 + x + 1 = 4 x 5 - 3x 2 + x + 1 The quotient is 2x 2 ; the remainder is -x 2 + x + 1. 5x 4 + 10x 2 -11x 2 + x - 2 -11x 2 - 22 x + 20 11. 2x 3 -1 4 x 5 - 3x 2 + x + 1 4 x 5 - 2x 2 - x2 + x +1 ) 2x 2 12. 3 3x -1 3x - x + x - 2 5 2 ) x2 3x 5 - x 2 x -2 Check : (3x 3 -1)( x 2 ) + ( x - 2) = 3x 5 - x 2 + x - 2 The quotient is x 2 ; the remainder is x - 2. 31 Chapter R 13. Review x 2 - 2x + Check : 2 5 1 1 2 x - 2x + (2x + x + 1)+ x + 2 2 2 = 2x 4 + x 3 + x 2 - 4 x 3 - 2x 2 - 2x + x 2 + 1 x 2 1 1 5 + + x+ 2 2 2 1 2 2 4 3 2x + x + 1 2x - 3x + ) x+ 1 + x2 2x 4 + x 3 - 4x3 - x2 + x -4 x 3 - 2x 2 - 2x x 2 + 3x + 1 1 1 x2 + x + 2 2 5 1 x+ 2 2 14. 2 1 x2 - x - 3 9 3x 2 + x + 1 3x 4 - x 3 + = 2x 4 - 3x 3 + x + 1 1 The quotient is x 2 - 2x + ; 2 5 1 the remainder is x + . 2 2 Check : ) x- 2 3x 4 + x 3 + x 2 - 2x 3 - x 2 + x 2 2 -2x 3 - x 2 - x 3 3 1 5 - x2 + x - 2 3 3 1 1 1 - x2 - x - 3 9 9 16 17 x- 9 9 1 16 17 2 (3x 2 + x + 1) x 2 - x - + x - 9 9 9 3 2 1 1 = 3x 4 - 2x 3 - x 2 + x 3 - x 2 - x 3 9 3 1 16 17 2 + x2 - x - + x - 9 9 9 3 4 3 = 3x - x + x - 2 2 1 The quotient is x 2 - x - ; 3 9 16 17 the remainder is x- . 9 9 15. x -1 - 4 x + x + 3 2 ) - 4 x 2 - 3x - 3 -4 -4x3 + 4x2 - 3x 2 -3x 2 + 3x - 3x - 4 -3x + 3 -7 Check : (x -1)(- 4 x 2 - 3x - 3) + (- 7) = - 4 x 3 - 3x 2 - 3x + 4 x 2 + 3x + 3 - 7 = -4x3 + x2 - 4 The quotient is - 4x 2 - 3x - 3; the remainder is 7. 32 Section R.6 16. x -1 - 3x Polynomial Division; Synthetic Division Check: (x - 1)(- 3x 3 - 3x 2 - 3x - 5) + (- 6 ) = -3x 4 - 3x 3 - 3x 2 - 5x + 3x 3 + 3x 2 + 3x + 5 - 6 = -3x 4 - 2x - 1 The quotient is - 3x 3 - 3x 2 - 3x - 5; the remainder is 6. ) - 3x 3 - 3x 2 - 3x - 5 4 4 3 - 2x -1 - 3x 3 -3x 3 + 3x 2 - 3x 2 - 2x -3x 2 + 3x - 5x -1 -5x + 5 -6 - 3x + 3x 17. x2 + x +1 x4 ) x 2 - x -1 - x2 - x 3 - 2x 2 -x 3 - x 2 - x - x2 + x + 1 -x 2 - x -1 2x + 2 +1 x4 + x3 + x2 Check : (x 2 + x + 1)( x 2 - x -1) + (2x + 2) = x 4 - x 3 - x 2 + x 3 - x 2 - x + x 2 - x -1+ 2x + 2 = x4 - x2 +1 The quotient is x 2 - x - 1; the remainder is 2x + 2. 18. x2 - x +1 x4 4 3 ) x 2 + x -1 - x2 2 +1 x - x +x x 3 - 2x 2 x3 - x2 + x - x2 - x + 1 -x 2 + x -1 - 2x + 2 Check: (x 2 - x + 1)(x 2 + x - 1) + (- 2x + 2) = x 4 + x 3 - x 2 - x 3 - x 2 + x + x 2 + x - 1- 2x + 2 = x4 - x 2 + 1 The quotient is x 2 + x - 1; the remainder is - 2x + 2 . 33 Chapter R 19. x-a x Review x 2 + ax + a 2 3 ) Check : -a 3 (x - a)(x 2 + ax + a 2 ) + 0 = x 3 + ax 2 + a 2 x - ax 2 - a 2 x - a 3 = x 3 - a3 x 3 - ax 2 ax 2 ax 2 - a 2 x a x-a 2 3 The quotient is x 2 + ax + a 2 ; the remainder is 0. a2 x - a3 0 20. x 4 + ax 3 + a 2 x 2 + a 3 x + a 4 - a5 ax 4 ax 4 - a 2 x 3 a2 x 3 a2 x 3 - a3 x 2 a3 x 2 a3 x 2 - a4 x a4 x - a5 a4 x - a5 0 21. 2) 1 - 1 2 1 1 -1)1 2 4 2 8 4 12 Quotient: x 2 + x + 4 Remainder: 12 Quotient: x 2 + x - 4 Remainder: 5 Quotient: 3x 2 + 11x + 32 Remainder: 99 = x 5 - a5 The quotient is x 4 + ax 3 + a2 x 2 + a3 x + a 4 ; the remainder is 0. x 5 - ax 4 Check : (x - a)(x 4 + ax 3 + a 2 x 2 + a 3 x + a 4 ) + 0 = x 5 + ax 4 + a 2 x 3 + a 3 x 2 + a 4 x - ax 4 - a2 x 3 - a3 x 2 - a4 x - a5 x - a x5 ) 22. 2 -3 1 -1 -1 4 1 1 -4 5 3 96 99 23. 3) 3 2 -1 9 33 3 11 32 34 Section R.6 24. - 2) - 4 2 -1 1 8 - 20 42 - 4 10 - 21 43 0 -4 0 1 0 -3 9 -15 45 -138 5 -15 46 -138 Polynomial Division; Synthetic Division Quotient: - 4x 2 + 10x - 21 Remainder: 43 Quotient : x 4 - 3x 3 + 5x 2 -15x + 46 Remainder : -138 Quotient: x 3 + 2x 2 + 5x + 10 Remainder: 22 4 5 + 4x 4 + x 3 + x 2 + 2x + 2 x 25. -3)1 1 -3 26. 2) 1 0 1 0 2 2 4 10 20 1 2 5 10 22 1) 4 4 0 -3 4 4 4 1 0 1 1 1 0 1 2 2 2 5 2 7 27. Quotient : Remainder: 7 Quotient: x 4 - x 3 + 6x 2 - 6x + 6 Remainder: 16 28. -1)1 0 5 0 0 - 10 -1 1 - 6 6 - 6 1 - 1 6 - 6 6 - 16 29. -1.1 0.1 ) 0 0.2 - 0.11 0.121 0.1 - 0.11 0.321 0 - 0.2 - 0.21 0.441 0.1 - 0.21 0.241 0 - 0.3531 - 0.3531 Quotient: 0.1x 2 - 0.11x + 0.321 Remainder: 0.3531 Quotient: 0.1x - 0.21 Remainder: 0.241 Quotient: x 4 + x 3 + x 2 + x + 1 Remainder: 0 Quotient: x 4 - x 3 + x 2 - x + 1 Remainder: 0 30. - 2.1) 0.1 31. 1) 1 0 0 0 0 - 1 1 1 1 1 1 1 1 1 1 1 0 -1)1 0 0 0 0 1 -1 1 -1 1 -1 1 -1 1 -1 1 0 32. 35 Chapter R 33. 2) 4 - 3 - 8 8 10 4 5 2 -3) - 4 Review 4 4 8 Remainder = 8 0; therefore x - 2 is not a factor of the given polynomial. Remainder = 161 0; therefore x + 3 is not a factor of the given polynomial. Remainder = 0; therefore x - 2 is a factor of the given polynomial. Remainder = 0; therefore x - 2 is a factor of the given polynomial. 0 27 9 - 27 9 0 Remainder = 0; therefore x + 3 is a factor of the given polynomial. Remainder = 0; therefore x + 3 is a factor of the given polynomial. Remainder = 1 0; therefore x + 3 is not a factor of the given polynomial. Remainder = 0; therefore x + 4 is a factor of the given polynomial. Remainder = 0; 1 therefore x - is a factor of the 2 given polynomial. Remainder = 2 0; 1 therefore x + is not a factor of the 3 given polynomial. 34. 5 0 8 12 - 51 153 - 4 17 - 51 161 0 -5 0 0 0 -5 10 - 10 0 35. 2) 3 - 6 6 3 0 2) 4 0 8 4 8 36. - 15 0 - 4 16 2 4 1 2 0 37. -3) 3 0 0 82 0 - 9 27 - 81 - 3 3 - 9 27 1 -3 0 -6 2 -6 38. -3) 2 - 18 0 1 0 -9 18 0 0 - 3 9 0 0 1 -3 0 39. - 4) 4 0 - 64 0 1 0 - 15 - 16 64 0 0 - 4 16 4 - 16 0 0 1 -4 1 - 16 0 1 0 16 0 0 - 4 0 0 1 -4 - 16 16 0 40. - 4) 1 0 -4 1 -4 41. 1 ) 2 -1 0 2 - 1 2 1 0 0 1 2 0 0 2 0 42. - 1 )3 3 3 1 0 -3 1 -1 0 0 1 0 0 -3 2 36 Section R.6 43. Polynomial Division; Synthetic Division x 3 - 2x 2 + 3x + 5 d = ax 2 + bx + c + x +2 x +2 In order to find a + b + c + d , we do the long division and then look at the coefficients of the quotient and remainder. Divide: Therefore, x 2 - 4 x + 11 x 3 - 2x 2 + 3x + 5 -17 = x 2 - 4 x + 11+ 3 2 x+2 x+2 x + 2 x - 2x + 3x + 5 a + b + c + d = 1- 4 + 11-17 x 3 + 2x 2 = -9 - 4 x 2 + 3x ) -4 x 2 - 8x 11x + 5 11x + 22 -17 44. Answers will vary. 37 ...
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This homework help was uploaded on 04/07/2008 for the course MAC 1105 taught by Professor Any during the Spring '08 term at FIU.

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