Skills Practice Skills Practice for Lesson 10.1

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1 Skills Practice Skills Practice for Lesson.1 Name Date Higher Order Polynomials and Factoring Roots of Polynomial Equations Problem Set Solve each polynomial equation using factoring. Then check your solution(s). 1. x 2 2x x 2 4x 4 x 2 2x 15 0 (x 3)(x 5) 0 x 5, 3 Check: ( 5) 2 2( 5) (3) 2 2(3) Chapter l Skills Practice 663

2 3. 0 4x 2 4x x x 2 7x x Chapter l Skills Practice

3 Name Date 7. x 3 11x 2 28x 0 8. x 4 12x 3 36x 2 0 Chapter l Skills Practice 665

4 9. x 3 2x 2 x 2. x 3 3x 2 4x Chapter l Skills Practice

5 Name Date 11. x 4 9x x 5 x 3 21x Chapter l Skills Practice 667

6 13. 2x 3 18x x Chapter l Skills Practice

7 Name Date 15. 3x 3 6 2x 2 9x 16. x 3 3x 2 3x 1 0 Chapter l Skills Practice 669

8 Solve each polynomial equation. Then check your solution(s). 17. x 2 6x 2 0 x b b 2 4ac 2a ( 6) ( 6) x 2 4(1)(2) 2(1) x x 3 7 Check: ( 3 7 ) 2 6 ( 3 7 ) 2 0 ( 3 7 ) 2 6 ( 3 7 ) 2 0 ( ) ( ) x 3 8x 2 4x Chapter l Skills Practice

9 Name Date 19. x 3 4x 2 5x x 4 2x x 2 2x 2 0 Chapter l Skills Practice 671

10 22. x 5 2x 4 7x 3 14x 2 18x Chapter l Skills Practice

11 Name Date 23. 4x 4 24x 3 40x 2 0 Chapter l Skills Practice 673

12 24. x 3 7x 5x Chapter l Skills Practice

13 Name Date x 5 2 x x 0 Chapter l Skills Practice 675

14 676 Chapter l Skills Practice

15 Skills Practice Skills Practice for Lesson.2 Name Date Greater Than or Less Than Solving Polynomial Inequalities Problem Set Write the compound inequalities needed to solve each inequality. Do not solve the compound inequalities. 1. (x 3)(x 7) 0 Case 1: x 3 0 and x 7 0 Case 2: x 3 0 and x x(x 4)(x 6) 0 3. x 2 3x 0 Chapter l Skills Practice 677

16 4. x 3 5x 2 14x 0 5. x 3 4x 2 9x (x 2)(x 3)(x 4)(x 1) Chapter l Skills Practice

17 Name Date Solve each case for each compound inequality. 7. Case 1: x 2 0 and x 5 0 x 2 and x 5 x 2 Case 2: x 2 0 and x 5 0 x 2 and x 5 x 5 8. Case 1: x 3 0 and x 8 0 Case 2: x 3 0 and x Case 1: x 0 and x 2 0 and x 6 0 Case 2: x 0 and x 2 0 and x 6 0 Case 3: x 0 and x 2 0 and x 6 0 Case 4: x 0 and x 2 0 and x 6 0 Chapter l Skills Practice 679

18 . Case 1: x 0 and x 2 0 and x 7 0 Case 2: x 0 and x 2 0 and x 7 0 Case 3: x 0 and x 2 0 and x 7 0 Case 4: x 0 and x 2 0 and x Chapter l Skills Practice

19 Name Date 11. Case 1: x 2 0 and x 3 0 and x 4 0 and x 1 0 Case 2: x 2 0 and x 3 0 and x 4 0 and x 1 0 Case 3: x 2 0 and x 3 0 and x 4 0 and x 1 0 Case 4: x 2 0 and x 3 0 and x 4 0 and x 1 0 Case 5: x 2 0 and x 3 0 and x 4 0 and x 1 0 Case 6: x 2 0 and x 3 0 and x 4 0 and x 1 0 Case 7: x 2 0 and x 3 0 and x 4 0 and x 1 0 Case 8: x 2 0 and x 3 0 and x 4 0 and x 1 0 Chapter l Skills Practice 681

20 12. Case 1: x 1 0 and x 2 0 and x 3 0 Case 2: x 1 0 and x 2 0 and x 3 0 Case 3: x 1 0 and x 2 0 and x 3 0 Case 4: x 1 0 and x 2 0 and x 3 0 Solve each inequality algebraically. 13. (x 4)(x 7) 0 Case 1: x 4 0 and x 7 0 x 4 and x 7 Case 2: x 4 0 and x 7 0 x 4 and x 7 7 x 4 Check: ( 8 4)( 8 7) ( 12)( 1) 12 0 (0 4)(0 7) ( 4)(7) 28 0 (5 4)(5 7) (1)(12) 12 0 The solution is 7 x Chapter l Skills Practice

21 Name Date 14. x 2 2x 48 0 Chapter l Skills Practice 683

22 15. (x 2)(x 4)(x 7) Chapter l Skills Practice

23 Name Date 16. x 3 x 2 16x 16 0 Chapter l Skills Practice 685

24 17. 4x 3 0x Chapter l Skills Practice

25 Name Date 18. x 4 5x Chapter l Skills Practice 687

26 Use the graph of f( x) to solve each inequality. 19. f(x) 1 2 x2 1 2 x f(x) x3 x 2 12x y y f (x) 8 4 f (x) x x Solve: 1 2 x2 1 2 x 3 0. Solve: x3 x 2 12x 0. The solution is 3 x f(x) 1 2 x2 3 x f(x) x3 2 y y f (x) x f (x) x Solve: 1 2 x2 3 2 x 2 0. Solve: x Chapter l Skills Practice

27 Name Date 23. f(x) x 4 4x 3 4x 2 16x 24. f(x) x 2 9 y y f (x) f (x) x x Solve: x 4 4x 3 4x 2 16x 0. Solve: x Chapter l Skills Practice 689

28 Solve each polynomial inequality graphically. 25. x 2 x x 2 x 12 0 x 2 x 6 0 (x 2)(x 3) 0 x 2, 3 Check: ( 2) 2 ( 2) (3) y x The solution is x 2 or x 3. Check: ( 3) 2 ( 3) (0) (4) Chapter l Skills Practice

29 Name Date 27. x 3 x 2 6x x 3 2x 2 x 2 0 Chapter l Skills Practice 691

30 29. x x Chapter l Skills Practice

31 Skills Practice Skills Practice for Lesson.3 Name Date It s Fundamental The Fundamental Theorem of Algebra Vocabulary Write the term(s) from the box that best completes each statement. The terms may be used more than once. Fundamental Theorem of Algebra double roots complex roots triple roots 1. The states that any polynomial equation of degree n must have exactly n. 2. The roots 5 and 8 of the function f( x) ( x 5) 2 (x 8) 2 are. 3. When of a polynomial equation appear more than once, they have a multiplicity greater than have a multiplicity of The of a polynomial function f(x) are those values of x such that f(x) 0. Problem Set Use the Fundamental Theorem of Algebra to determine the number of complex roots of each polynomial equation. 1. x 2 2x x 2 x 8 0 The equation has 2 complex roots x x x 2 3x x x x 2 2x x 5 2x 4 3x 3 9x 2 3x x 2 x 3 4x 5 0 Chapter l Skills Practice 693

32 The roots of a polynomial equation are given. Determine the multiplicity of the roots. 9. The roots of a second-degree equation are x 1 and x 1. The root 1 has a multiplicity of one. The root 1 has a multiplicity of one.. The roots of a second-degree equation are x 2 and x The roots of a second-degree equation are x 5 and x The roots of a third-degree equation are x 2, x 2, and x The roots of a third-degree equation are x 0, x 3, and x The roots of a fourth-degree equation are x 4, x 4, x 2, and x The roots of a fourth-degree equation are x 3, x 6, x 6, and x The roots of a fifth-degree equation are x 2, x 2, x 0, x 0, and x 0. For each polynomial equation, state the number of complex roots and use any method to calculate the roots. State the multiplicity of the roots. Then check your solution(s). 17. x 2 3x 2 0 (x 2)(x 1) 0 x 2, 1 There are two complex roots, each with a multiplicity of one. Check: x 2 3x 2 0 ( 2) 2 3( 2) ( 1) 2 3( 1) Chapter l Skills Practice

33 Name Date 18. x x 3 3x 2 0x Chapter l Skills Practice 695

34 20. x 4 68x x 4 2x Chapter l Skills Practice

35 Name Date 22. x 3 5x 2 x x 4 4x 3 4x 2 0 Chapter l Skills Practice 697

36 24. x 3 x 2 25x 0 For each polynomial function, state the number of complex zeros and use any method to calculate the zeros. State the multiplicity of the zeros. Then check your solution(s). 25. f(x) x 3 There are three complex zeros all equal to 0. The zero 0 has a multiplicity of three. Check: (0) f(x) (x 3) Chapter l Skills Practice

37 Name Date 27. f(x) x 5 x 4 9x 3 9x f(x) x 5 99x 3 0x Chapter l Skills Practice 699

38 29. f(x) x 5 3x 3 5x 30. f(x) x 4 6x Chapter l Skills Practice

39 Skills Practice Skills Practice for Lesson.4 Name Date When Division is Substitution Polynomial and Synthetic Division Vocabulary Define the given term in your own words. 1. Synthetic division Problem Set For each of the following, q(x) is the quotient and r(x) is the remainder when f(x) is divided by p(x). Write the dividend as the product of the divisor and the quotient plus the remainder. 1. f(x) 3x 3 8x 2 5x 7 2. f(x) x 2 3 p(x) x 2 p(x) x 3 q(x) 3x 2 2x 1 q(x) x 3 r(x) 9 r(x) 12 f(x) (x 2) ( (3x 2 2x 1) 9 x 2 ) 3. f(x) x 4 5x 3 4x f(x) x 3 5x 2 2 p(x) x 1 p(x) x 4 q(x) x 3 5x 2 9x 9 q(x) x 2 x 4 r(x) 0 r(x) 18 Chapter l Skills Practice 701

40 5. f(x) x 3 2x 2 23x f(x) 4x 4 6x 3 8x 2 4x p(x) x 3 p(x) 2x 1 q(x) x 2 x 20 r(x) 0 q(x) 2x 3 2x 2 3x r(x) x Calculate each quotient using long division. 7. x ) 5x 4 0x 3 2x 2 8. x 1 ) x 3 6x 2 11x 6 5x 3 0x 2 2x x ) 5x 4 0x 3 2x 2 5x 4 0x 3 2x 2 ( 2x 2 ) 0 9. x 6 ) 4x 4 22x 3 12x 2. x 5 ) 2x 3 x 3 3x x 1 ) 6x 4 17x 3 8x 2 8x x 5 ) 6x 4 19x 3 18x 2 x Chapter l Skills Practice

41 Name Date Explain the mistake made in setting up the process of synthetic division to divide f( x) by p( x). 13. f( x) x 4 3x 2 x f( x) 3x 3 7x 2 5x 3 p( x) x 2 p( x) x The x 3 term is missing. Zero must be used as a placeholder. 15. f( x) x 4 6x 3 2x 2 x f( x) x 3 5x 2 7x p( x) 2x 5 p( x) x f( x) x 3 4x 2 7x f( x) 4x 3 x 2 x 9 p( x) x 8 p( x) 2x Calculate each quotient using synthetic division. 19. f(x) x 3 6x 2 5x 12 divided by x x 2 3x 4 Chapter l Skills Practice 703

42 20. f(x) x 3 5x 2 18x 8 divided by x f(x) 2x 3 3x 2 19x 15 divided by 2x f(x) 4x 3 23x 2 39x 18 divided by 4x x 4 3x 3 6x divided by x Chapter l Skills Practice

43 Name Date 24. x 3 2x 2 8x 14 divided by x x 4 19x 3 41x 2 43x 60 divided by 2x x 5 17x 4 6x 3 35x 2 6x 12 divided by 7x 3 Chapter l Skills Practice 705

44 706 Chapter l Skills Practice

45 Skills Practice Skills Practice for Lesson.5 Name Date Remains of a Polynomial The Remainder Theorem Vocabulary Write the term that best completes each statement. 1. The result of using synthetic division to evaluate a polynomial function for a specific value r is called because f(r) R, the remainder. 2. The states that when any polynomial equation or function is divided by a linear factor (x r), the remainder is the value of the equation or function when x r. Problem Set Determine the remainder of each function by evaluating it at the given value. 1. 2x 4 x 3 x 2 11x 15 divided by x 3 2(3) 4 (3) 3 (3) 2 11(3) x 3 6x 2 4x 1 divided by 3x 2 Chapter l Skills Practice 707

46 3. 4x 4 6x 3 x 2 divided by 2x x 4 7x 3 11x 2 12x 40 divided by x x 3 11x 2 20x 20 divided by x x 3 3x 2 6x 2 divided by x Chapter l Skills Practice

47 Name Date 7. 3x 4 4x 2 2x 1 divided by x 3 8. x 5 4x 3 x 2 divided by x 4 If f(r) R, determine the remainder when f( x) is divided by x r for the given values of r and R. 9. r 4 and R 11. r 1 and R 7 The remainder is r 12 and R r 16 and R r 1 and R r 7 and R r 2 and R r 9 and R 0 Chapter l Skills Practice 709

48 Use synthetic substitution to evaluate each polynomial function for the given value of r. 17. f(x) 2x 5 3x 4 x 3 2x 6, r x 3 3x 2 4x 3 divided by x f( 3) x 4 5x 3 2x divided by x x 3 5x 2 4x 4 divided by x f( x) 2x 3 4x 2 9, r f( x) x 4 3x 3 4x 2 x 11, r f( x) x 5 2x 4 x 3 x 2 7x 1, r f( x) 5x 3 3x 2 x 2, r 2 7 Chapter l Skills Practice

49 Name Date Use synthetic substitution to find the remainder when f( x) is divided by p( x). 25. f( x) 5x 3 3x 2 2x f( x) x 4 2x 3 x 2 x p( x) x 2 p( x) x The remainder is f( x) x 3 4x 2 x f( x) 2x 5 x 4 2x 3 x 2 x 4 p( x) x 3 p( x) x f( x) 4x 2 6x f( x) 3x 3 x 2 x 3 p( x) 2x 1 p( x) 3x f( x) x f( x) x 4 3x 2 x 3 p( x) 2x 1 p( x) 3x 1 Chapter l Skills Practice 711

50 712 Chapter l Skills Practice

51 Skills Practice Skills Practice for Lesson.6 Name Date More Factors of a Polynomial The Factor Theorem Vocabulary Define the term in your own words. 1. Factor Theorem Problem Set Determine whether the second polynomial is a factor of the first. 1. 2x 4 7x 3 11x 2 12x 40, x x 3 11x 2 20x 20, x 2 Chapter l Skills Practice 713

52 3. 5x 3 3x 2 6x 2, x x 4 4x 3 8x 5, x x 3 27x 2 44x 40, 3x x 5 13x 4 5x 3 6x 2, 3x Chapter l Skills Practice

53 Name Date Determine whether the second polynomial expression is a factor of the first. If it is, completely factor the first polynomial. 7. x 3 1, x 1 8. x 4 16, x 2 Chapter l Skills Practice 715

54 9. 3x 4 2x 3 5, x 2. 5x 3 8x 2 9x 12, x Chapter l Skills Practice

55 Name Date 11. x 4 x 3 13x 2 25x 300; x 3, x x 4 x 3 3x 2 5x ; x 1, x 2 Chapter l Skills Practice 717

56 Determine an equation that would have each set of roots. 13. x 2, 5, 3 (x 2)(x 5)(x 3) (x 2 3x )(x 3) x 3 3x 2 x 3x 2 9x 30 x 3 19x x 3, x 7 i, x 2, 2 3, 5 4 i 17. x 0, 4, 2 i 718 Chapter l Skills Practice

57 Name Date 18. x 2, 1 2, 3i Calculate the product and sum of each set of roots. 19. The roots of the equation x 4 7x are x 2, 3. Product: 2 ( 2) 3 ( 3 ) 12 Sum: 2 ( 2) 3 ( 3 ) The roots of the equation x 4 4x 3 x 2 16x 20 0 are x 5, 1, 2i. Product: Sum: 21. The roots of the equation 4x 3 9x 2 22x 5 are x 1, 1 2i. 4 Product: Sum: Chapter l Skills Practice 719

58 22. The roots of the equation x 4 15x 2 x 24 0 are x 3, 2, 1, 4. Product: Sum: 23. The roots of the equation x 3 6x 2 6x 36 are x 6, 6. Product: Sum: 24. The roots of the equation 9x 4 3x 3 30x 2 6x 12 0 are x 2, 1, Product: Sum: Without finding the roots, calculate the product and sum of the roots for each polynomial equation x 3 x 2 32x x 3 25x 2 154x 40 0 Product: Sum: x 4 3x 3 x x 4 2x 2 7x x 6 x 4 31x x 5 x 4 7x 3 7x 2 12x Chapter l Skills Practice

59 Skills Practice Skills Practice for Lesson.7 Name Date Let s Be Rational Rational Root Theorem Vocabulary Define the term in your own words. 1. Rational Root Theorem Problem Set List all possible rational roots for each polynomial equation. 1. x 4 2x 3 x x 3 5x 2 7x , 1 3. x 5 7x 4 2x 3 x 2 x x 2 6x x 5 2x x 6 9x 5 6x 3 2x x 2 9x 3 7x x 5 x 4 2x 2 0 Chapter l Skills Practice 721

60 The number in each exercise is one root of a polynomial equation. Identify a second root i i i i i List the possible rational roots of each polynomial equation. Then use synthetic division to determine one of the rational roots. 17. x 2 x x 2 8x , One rational root is Chapter l Skills Practice

61 Name Date 19. x 3 x 2 8x x 3 x 2 31x x x 3 5x 2 9x 45 0 Chapter l Skills Practice 723

62 Determine all roots for each polynomial equation. 23. x 2 2x 15 0 Possible rational roots: 1, 3, 5, 15 Use synthetic division to determine one of the roots Rewrite the original polynomial as a product. x 2 2x 15 ( x 3 ) ( x 5 ) The roots of the original polynomial equation are 3 and x 2 7x Chapter l Skills Practice

63 Name Date 25. x 3 3x 2 4x 12 0 Chapter l Skills Practice 725

64 26. 2x 4 3x 3 4x 2 9x Chapter l Skills Practice

65 Name Date 27. 2x 5 3x 4 12x 3 18x 2 14x 21 0 Chapter l Skills Practice 727

66 28. x 4 5x 3 x 2 9x Chapter l Skills Practice

67 Skills Practice Skills Practice for Lesson.8 Name Date Getting to the Root of it All Solving Polynomial Equations Problem Set Use the Fundamental Theorem of Algebra to determine how many complex roots exist for each polynomial equation. 1. x 4 2x 3 5x 2 6x x 3 1 4x 5 4x x x 4 x 2 4x x 4 6x 3 2x 5 x 2 8x x 6 2x 4 5x Each set of numbers represents all roots of a polynomial equation. Classify each root as rational, irrational, imaginary, or complex rational rational i i Chapter l Skills Practice 729

68 i 2 2 i 8 11 i 2 i Determine all roots of each polynomial equation. Then classify each root as rational, irrational, imaginary, or complex. 13. x 3 6x 2 9x Possible rational roots: 1, 2, 7, 14 Use synthetic division to determine one of the roots Rewrite the original polynomial as a product. x 3 6x 2 9x 14 (x 1)(x 2 5x 14) Factor x 2 5x 14. x 2 5x 14 (x 7)(x 2) Rewrite the original polynomial as a product. x 3 6x 2 9x 14 (x 1)(x 7)(x 2) Solutions: x 2, 1, 7 Root 1: x 2, rational Root 2: x 1, rational Root 3: x 7, rational 730 Chapter l Skills Practice

69 Name Date 14. x 3 2x 2 2x 4 0 Chapter l Skills Practice 731

70 15. x 3 3x 2 3x Chapter l Skills Practice

71 Name Date 16. x 4 3x Chapter l Skills Practice 733

72 17. 2x 4 13x 3 30x 2 28x Chapter l Skills Practice

73 Name Date 18. x 3 5x 2 9x 5 0 Chapter l Skills Practice 735

74 19. x 4 8x 3 22x 2 24x Chapter l Skills Practice

75 Name Date 20. x 4 3x Chapter l Skills Practice 737

76 738 Chapter l Skills Practice

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