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1 1. Dividing: 4x3! 8x 2 + 6x 2x 5.7 Division of Polynomials = 4x3 2x! 8x2 2x + 6x 2x = 2x2! 4 3. Dividing: 1x4 + 15x 3! 2x 2!5x 2 = 1x4!5x x3!5x 2! 2x2!5x 2 =!2x2! 3x Dividing: 8y5 + 1y 3! 6y 4y 3 = 8y5 4y + 1y3 3 4y! 6y 3 4y = 3 2y ! 3 2y 2 7. Dividing: 5x3! 8x 2! 6x = 5x3!2x 2!2x! 8x2 2!2x! 6x 2!2x =! x x 9. Dividing: 28a3 b a 4 b 3 = 28a3 b a4 b 3 = 4ab 3 + 6a 2 b 7a 2 b 2 7a 2 b 2 7a 2 b Dividing: 1x3 y 2! 2x 2 y 3! 3x 3 y 3!1x 2 y = 1x3 y 2!1x 2 y! 2x2 y 3!1x 2 y! 3x3 y 3!1x 2 y =!xy + 2y2 + 3xy 2 ( x + 2) 13. Dividing by factoring: x2! x! 6 = x! 3 = x + 2 x! 3 x! Dividing by factoring: 2a2! 3a! 9 = ( 2a + 3) ( a! 3) = a! 3 2a + 3 2a Dividing by factoring: 5x2! 14xy! 24y 2 5x + 6y = x! 4y x! 4y 19. Dividing by factoring: x3! y 3 x! y = x! y x2 + xy + y Dividing by factoring: y4! 16 y! 2 = y y! Dividing by factoring: x 3 + 2x 2! 25 = x2 x + 2 x! 4y x! y y 2! 4! 25( x + 2) x2! 25 = x + 2 = ( x + 2) x + 5 = x + 2 ( ) ( x + 5) = 5x + 6y = x 2 + xy + y 2 y + 2 = y2 + 4 ( y! 2) y! 2 = ( y 2 + 4) ( y + 2) Copyright 21 MathTV.com, Inc.

2 25. Dividing by factoring: 4x x 2! 9x! 27! 9( ) = 4x2 4x2! 9 = = ( ) 2 = 2 ( 2x! 3) ( 2x! 3) 27. Dividing using long division: 29. Dividing using long division: x! 7 2x + 5 x + 2 x 2! 5x! 7 3x! 4 6x 2 + 7x! 18 x 2 + 2x!7x! 7!7x! 14 6x 2! 8x 15x! 18 15x! The quotient is x! x + 2. The quotient is 2x x! Dividing using long division: 33. Dividing using long division: 2x 2! 5x + 1 y 2! 3y! 13 x + 1 2x 3! 3x 2! 4x + 5 2y! 3 2y 3! 9y 2! 17y x 3 + 2x 2!5x 2! 4x!5x 2! 5x x + 5 x y 3! 3y 2!6y 2! 17y!6y 2 + 9y!26y + 39!26y + 39 The quotient is 2x 2! 5x x + 1. The quotient is y2! 3y! 13. Copyright 21 MathTV.com, Inc.

3 35. Dividing using long division: 37. Dividing using long division: 3y 2 + 6y + 8 x! 3 2x 2! 3x + 2 2x 3! 9x x! 6 2x 3! 3x 2 + 2x!6x 2 + 9x! 6!6x 2 + 9x! 6 2y! 4 6y 3 + y 2! 8y + 5 6y 3! 12y 2 12y 2! 8y 12y 2! 24y 16y y! The quotient is x! 3. The quotient is 3y 2 + 6y y! Dividing using long division: 41. Dividing using long division: a 3 + 2a 2 + 4a + 6 a! 2 a 4 + a 3 + a 2! 2a + 5 a 4! 2a 3 2a 3 + a 2 2a 3! 4a 2 4a 2! 2a 4a 2! 8a y 3 + 2y 2 + 4y + 8 y! 2 y 4 + y 3 + y 2 + y! 16 y 4! 2y 3 2y 3 + y 2 2y 3! 4y 2 4y 2 + y 4y 2! 8y 6a + 5 8y! 16 6a! 12 8y! The quotient is a 3 + 2a 2 + 4a a! 2. The quotient is y3 + 2y 2 + 4y + 8. Copyright 21 MathTV.com, Inc.

4 43. Dividing using long division: x 2! 2x + 1 x 2 + 3x + 2 x 4 + x 3! 3x 2! x + 2 x 4 + 3x 3 + 2x 2!2x 3! 5x 2! x!2x 3! 6x 2! 4x x 2 + 3x + 2 x 2 + 3x + 2 The quotient is x 2! 2x First use long division to find the remaining factors: x 2 + 3x + 2 x 3 + 6x x + 6 x 3 + 3x 2 3x x 3x 2 + 9x 2x + 6 2x + 6 Thus x 3 + 6x x + 6 = = x 2 + 3x First use long division to find the remaining factors: x 2 + 2x! 8 x 3 + 5x 2! 2x! 24 x 3 + 3x 2 2x 2! 2x 2x 2 + 6x!8x! 24!8x! 24 Thus x 3 + 5x 2! 2x! 24 = 49. Yes, both answers are identical. = x 2 + 2x! 8 ( x + 2) x + 1. ( x + 4) ( x! 2). Copyright 21 MathTV.com, Inc.

5 x 4! 5x 3!x 2 + 6x Intermediate Algebra = (!2) 2! 5 (!2)! 7 = 4 + 1! 7 = P!2 The answer is the same. 53. a. Using long division: x 2! x! 2 x 3! 3x 2 + 5x! 6 x 3! 2x 2!x 2 + 5x!x 2 + 2x 3x! 6 3x! 6 Since the remainder is, x! 2 is a factor of x 3! 3x 2 + 5x! 6. Also note that: P( 2) = ( 2) 3! 3( 2) b. Using long division: x 3! x + 1 x 4! 5x 3! x 2 + 6! 6 = 8! ! 6 =!x 2 + 5x Since the remainder is, is a factor of x 4! 5x 3! x Also note that: P a. Completing the table: = ( 5) 4! 5( 5) 3! ( 5) 2 + 6( 5)! 5 = 625! 625! ! 5 = x C(x) b. The average cost function is C(x) = 2 x c. Completing the table: x C(x) d. The average cost function decreases. e. For y = C(x), the domain is x 1! x! 2 For y = C(x), the domain is x 1! x! 2 { } and the range is { y 2.15! y! 5.}. { } and the range is { y.25! y! 2.15}. Copyright 21 MathTV.com, Inc.

6 57. a. Substituting the values: T (1) = T (4) = T (5) = = = = b. The average cost function is T (m) = 4.95 m +.7. c. Substituting the values: T (1) = T (4) = T (5) = Performing the operations: +.7 =.1195 per minute +.7 =.824 per minute +.7 =.799 per minute 2a + 1 a 3 a 2 3a + 15 = 2( a + 5) a Performing the operations: x 2 " x + 2% (! 9) & ' = 63. Performing the operations: 2x! 7 x! 2! x! 2 = 2x! 7! x + 5 x! 2 1 % 65. Simplifying the expression: 1 x! x + 1 = Solving the equation: x x! = 3 x! 3 " x 2( x! 3) x! % 2& ' = 2 x! 3 = 6 2 x! 3 2x! 9 = 6 5x = 15 x = 3 There is no solution (3 does not check). " " x! 1 3& ' 3x 1 x + 1 % 3& ' 3x " 3 % x! 3& ' ( does not check) ( x! 3) = 3! x 3 + x a 2 3 a + 5 = 2 3a " x + 2% & ' = x! 3 = x! 2 x! 2 = 1 ( x + 2) Copyright 21 MathTV.com, Inc.

3.3 Dividing Polynomials Copyright Cengage Learning. All rights reserved. Objectives Long Division of Polynomials Synthetic Division The Remainder and Factor Theorems 2 Dividing Polynomials In this section

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