VII. FACTORISATION OVER C-, R- AND J-FIELDS
Use the factor theorem to show (1-4) that:
1. is a factor of
2. is a factor of
3. is a factor of
4. is a factor of
5. Given that , find the value of a if
6. Given that , find the value of a if .
7. Given that , find the values of a and b if i and are zeros of P(x).
8. Given that , find the values of a and b if P(2)=0 and P(1-i)=0.
Complete factorise the following (9-24):
9. over C
10. over C
11. over (i) J (ii) C
12. over C.
13. over C.
14. over (i) J and (ii) C.
15. over (i) Q (ii) R (iii) C.
16. over (i) R (ii) C
17. over (i) R (ii) C.
18. over C.
19. over (i) R (ii) C.
20. over (i) R (ii) C.
21. over (i) R (ii) C given that is one linear factor.
22. over C.
23. over (i) R (ii) C.
24. over C.
Find z in (25-31);
26. if (i) (ii)
32. Find the values of the real numbers a and b , such that 1+i is a root of the equation
Find z in 33-39:
Write down an equation of the lowest possible degree with (i) complex coefficients, (ii) rational coefficients and having the following among its roots:
40. 41. 42. ,
43. of multiplicity 2.
Find and ploton the complex plane (46-55):
46. The square root of 47. The square root of
50. 51. The cube root of 64.
52. The fourth root of 8. 53. The sixth root of -1
54. The sixth root of 64. 55. The cube root of -8.
56. Find real values of a for which ai is a solution of the polynomial equation
57. Find the values which the real numbers a and b must take for z=1 to be a root of the complex equation
58. If 1, , and are the cube roots of unity, prove that (i) , (ii) and (iii) .
59. If , prove that
Show that the roots of are the four complex roots of . Deduce that
60. Find (a) the three roots of the equation
(b) the remainder when is divided by
61. Find the real numbers k such that is a root of the equation . Hence or otherwise find the three roots of the equation.
62. Solve the following equations using a calculus method.
(a) given that it has a root of multiplicity 2.
(b) given that it has a root of multiplicity 3.