1. Chapter 6 More about Polynomials
Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Example 7
Example 8
Example 9
Example 10
Additional Example 6.1
Additional Example 6.2
Additional Example 6.3
Additional Example 6.4
Additional Example 6.5
Additional Example 6.6
Additional Example 6.7
Additional Example 6.8
Additional Example 6.9
Additional Example 6.10
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2. Chapter 6 More about Polynomials
Example 11
Example 12
Example 13
Example 14
Example 15
Example 16
Example 17
Additional Example 6.11
Additional Example 6.12
Additional Example 6.13
Additional Example 6.14
Additional Example 6.15
Additional Example 6.16
Additional Example 6.17
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3. Additional Example

4. Simplify the following.
Additional Example 6.1
Simplify the following.
Solution:

5. Additional Example 6.1
Solution:

6. Additional Example

7. Simplify the following.
Additional Example 6.2
Simplify the following.
Solution:

8. Additional Example 6.2
Solution:

9. Additional Example

10. Additional Example 6.3
Expand the following.
Solution:

11. Additional Example 6.3
Solution:

12. Additional Example

13. Additional Example 6.4
Find the quotient and the remainder of each of the following divisions.
Solution:
(a)

14. Additional Example 6.4
(b)
Solution:

15. Additional Example

16. Additional Example 6.5
Find the quotient and the remainder of each of the following divisions.
Solution:
(a)

17. Additional Example 6.5
(b)
Solution:

18. Additional Example

19. Additional Example 6.6
Find the quotient and the remainder of each of the following divisions.
(a)
Solution:

20. Additional Example 6.6
(b)
Solution:

21. Additional Example

22. Additional Example 6.7
When is divided by , the quotient is x + 6 and the remainder is 3x  24. Find the values of p and q.
Solution:

23. Additional Example

24. Additional Example

25. Find the remainder when is divided by
Additional Example 6.8
Find the remainder when is divided by
Solution:
(a) By the remainder theorem,

26. (b) By the remainder theorem, Solution:
Additional Example 6.8
(b) By the remainder theorem,
Solution:
(c) By the remainder theorem,

27. Additional Example

28. When is divided by , the remainder is 1. Find the value of k.
Additional Example 6.9
When is divided by , the remainder is 1. Find the value of k.
Solution:
By the remainder theorem,

29. Additional Example

30. Additional Example

31. When f (x) is divided by x + 3, the remainder is -170. Solution:
Additional Example 6.10
When is divided by x + 3 and 2x  1, the remainders are 170 and 16 respectively. Find the values of p and q.
When f (x) is divided by x + 3, the remainder is -170.
Solution:

32. When f (x) is divided by 2x - 1, the remainder is -16.
Additional Example 6.10
When f (x) is divided by 2x - 1, the remainder is -16.
Substitute p = -8 into (2),
Solution:

33. Additional Example

34. Additional Example

35. Additional Example 6.11
Let Use the factor theorem to determine whether each of the following is a factor of h(x).
(a) x  1
(b) x + 2
(c) 2x + 3
(d) 3x  2
Solution:

36. Additional Example 6.11
Solution:

37. Additional Example

38. Additional Example

39. Consider the polynomial (a) Prove that 2x  5 is a factor of g(x).
Additional Example 6.12
Consider the polynomial  
(a) Prove that 2x  5 is a factor of g(x).
(b) Factorize g(x).
(c) Solve the equation g(x) = 0.
Solution:

40. Additional Example 6.12
(b) By long division,
Solution:

41. Additional Example 6.12
Solution:

42. Additional Example

43. Additional Example

44. The polynomial is divisible by 2x + 3. When it is divided by
Additional Example 6.13
The polynomial is divisible by 2x + 3. When it is divided by
x  1, the remainder is 15. Find the values of p and q.
Solution:
Since g(x) is divisible by 2x + 3, the remainder is 0.

45. When g(x) is divided by x - 1, the remainder is -15.
Additional Example 6.13
When g(x) is divided by x - 1, the remainder is -15.
Substitute p = 3 into (2),
Solution:

46. Additional Example

47. Additional Example

48. Additional Example

49. Let Q(x) be the quotient and cx + d be the remainder when
Additional Example 6.14
Without doing an actual division, find the remainder when is divided by
Solution:
When x = 5,
Let Q(x) be the quotient and cx + d be the remainder when
is divided by (x - 5)(x + 3).
When x = -3,

50. Additional Example 6.14
Substitute c = 35 into (1),
Solution:

51. Additional Example

52. Additional Example

53. Factorize the following polynomials.
Additional Example 6.15
Factorize the following polynomials.
Solution:
[ The coefficient of x3 is 1 and the factors of the constant term 18 are 1, 2, 3, 6, 9, 18.
 The possible linear factors of f (x) are x  1, x  2, x  3, x  6, x  9, x  18.]
By the factor theorem,
 x - 1 is not a factor of f (x).
 x + 1 is not a factor of f (x).
 x - 2 is a factor of f (x).

54. Solution: By long division,
Additional Example 6.15
By long division,
Solution:
[ The factors of the coefficient of x3 are 1, 2, 3, 6 and the constant term is 1.
 The possible linear factors of f (x) are x  1, 2x  1, 3x  1, 6x  1.]

55. Solution: By the factor theorem,  x - 1 is a factor of f (x).
Additional Example 6.15
By long division,
Solution:
By the factor theorem,
 x - 1 is a factor of f (x).

56. Additional Example

57. Additional Example

58. Factorize the following polynomials.
Additional Example 6.16
Factorize the following polynomials.
Solution:
[ The factors of the coefficient of x3 are 1, 3 and the factors of the constant term -8 are 1, 2, 4, 8.
 The possible linear factors of f (x) are x  1, x  2, x  4, x  8, 3x  1, 3x  2,
3x  4, 3x  8.]
By the factor theorem,
 x - 1 is not a factor of f (x).
 x + 1 is not a factor of f (x).

59.  x - 2 is not a factor of f (x).
Additional Example 6.16
 x - 2 is not a factor of f (x).
 x + 2 is a factor of f (x).
By long division,
Solution:

60.  x - 1 is not a factor of f (x).
Additional Example 6.16
 The factors of the coefficient of x3 are 1, 2 and the factors of the constant term are 1, 2, 4, 8, 16.
 The possible linear factors of f (x) are x  1, x  2, x  4, x  8 , x  16 , 2x  1.
By the factor theorem,
 x - 1 is not a factor of f (x).
 x + 1 is not a factor of f (x).
 x - 2 is not a factor of f (x).
 x + 2 is not a factor of f (x).
 x - 4 is a factor of f (x).
Solution:

61. Additional Example 6.16
By long division,
Solution:

62. Additional Example

63. Additional Example

64. Additional Example 6.17
Determine whether each of the following polynomials has a linear factor with integral coefficient and constant term.
Solution:
(a) [ The factors of the coefficient of x3 are 1, 2 and the factors of the constant term are 1, 3, 5, 15.
 The possible linear factors with integral coefficient and constant term of the polynomial are x  1, x  3, x  5, x  15, 2x  1, 2x  3, 2x  5, 2x  15.]
By the factor theorem,

65. Additional Example 6.17
Solution:

66. Additional Example 6.17
Solution:

67. Additional Example 6.17
(b)  The factors of the coefficient of x3 are 1, 3 and the constant term is 1.
 The possible linear factors with integral coefficient and constant term of the polynomial are x  1, 3x  1.
By the factor theorem,
Solution: