, 1. Let P(x) = x6−2x5−x4+x3+2x2+x−2(a) Determine whether
(x−2)is a factor of P(x). (2)(b) Find all the possible rational zeros of
P(x)by using the Rational Zeros Theorem. (2)(c) Solve P(x) = 0. (4)3.
Find a fourth-degree polynomial with integer coefficients that has
zeros 3iand −1, with −1azero of multiplicity 2. 3.Determine
without converting the form and leave your answer in the form
a+bi, where a,b∈R.(i)i21 (i+2)(i−3)(2)(ii)(1+2i)(3+i)−2+i(4)5.
Simplify the following complex number, without changing the
polar form and leave youranswer in polar
∠2π3 3 3∠3π2 4∠7π3(6 Let Z = −1− √ 3i (i) Write Z in a
polar form (2) (ii) Use De Moivre’s Theorem to determine Z 4 . (3)
Z and leave your answer in polar form with the angle in radians (a)
Z = 1−i √ 3 2 (5) 2, 5π 4 , 2,− 5π 4 , −2,− π 4 . (3) (b)
Convert into rectangular coordinates: −4,− 13π 6
𝑷𝒓𝒐𝒃𝒍𝒆𝒎 𝟏: 𝑷𝒐𝒍𝒚𝒏𝒐𝒎𝒊𝒂𝒍 𝑨𝒏𝒂𝒍𝒚𝒔𝒊𝒔
𝑮𝒊𝒗𝒆𝒏 𝑷𝒐𝒍𝒚𝒏𝒐𝒎𝒊𝒂𝒍:
𝑃(𝑥) = 𝑥6 − 2𝑥5 − 𝑥4 + 𝑥3 + 2𝑥2 + 𝑥 − 2𝑃(𝑥)
= 𝑥6 − 2𝑥5 − 𝑥4 + 𝑥3 + 2𝑥2 + 𝑥 − 2
(𝒂) 𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒆 𝒘𝒉𝒆𝒕𝒉𝒆𝒓 (𝒙 − 𝟐)(𝒙
− 𝟐) 𝒊𝒔 𝒂 𝒇𝒂𝒄𝒕𝒐𝒓 𝒐𝒇 𝑷(𝒙)𝑷(𝒙).
𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏:
(x−2)is a factor of P(x). (2)(b) Find all the possible rational zeros of
P(x)by using the Rational Zeros Theorem. (2)(c) Solve P(x) = 0. (4)3.
Find a fourth-degree polynomial with integer coefficients that has
zeros 3iand −1, with −1azero of multiplicity 2. 3.Determine
without converting the form and leave your answer in the form
a+bi, where a,b∈R.(i)i21 (i+2)(i−3)(2)(ii)(1+2i)(3+i)−2+i(4)5.
Simplify the following complex number, without changing the
polar form and leave youranswer in polar
∠2π3 3 3∠3π2 4∠7π3(6 Let Z = −1− √ 3i (i) Write Z in a
polar form (2) (ii) Use De Moivre’s Theorem to determine Z 4 . (3)
Z and leave your answer in polar form with the angle in radians (a)
Z = 1−i √ 3 2 (5) 2, 5π 4 , 2,− 5π 4 , −2,− π 4 . (3) (b)
Convert into rectangular coordinates: −4,− 13π 6
𝑷𝒓𝒐𝒃𝒍𝒆𝒎 𝟏: 𝑷𝒐𝒍𝒚𝒏𝒐𝒎𝒊𝒂𝒍 𝑨𝒏𝒂𝒍𝒚𝒔𝒊𝒔
𝑮𝒊𝒗𝒆𝒏 𝑷𝒐𝒍𝒚𝒏𝒐𝒎𝒊𝒂𝒍:
𝑃(𝑥) = 𝑥6 − 2𝑥5 − 𝑥4 + 𝑥3 + 2𝑥2 + 𝑥 − 2𝑃(𝑥)
= 𝑥6 − 2𝑥5 − 𝑥4 + 𝑥3 + 2𝑥2 + 𝑥 − 2
(𝒂) 𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒆 𝒘𝒉𝒆𝒕𝒉𝒆𝒓 (𝒙 − 𝟐)(𝒙
− 𝟐) 𝒊𝒔 𝒂 𝒇𝒂𝒄𝒕𝒐𝒓 𝒐𝒇 𝑷(𝒙)𝑷(𝒙).
𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏:
