Analysis and Approaches HL Practice Paper Book | Page 104

2 . You are asked to investigate the factorization of a complex polynomial equation of even degree .

4 2

( ii ) Hence , show that the roots of the equation u �u �1� 0 is

� 5� � � 5�

� � � � � � � � � cos �� � � isin��

� , cos� � � � isin� � � , cos � isin and � 6 � � 6 � � 6� � 6� 6 6 5�

5� cos � isin . 6 6

( iii ) Solve the quadratic equation z the form cos�� isin�

.

2n n

� z �1� 0, giving the roots in

( �( cos �isin ))( � cos ( � ) �isin ( � ) ) as a ( b ) ( i ) Express z � � z � � � �

quadratic expression of z , giving the answer in terms of z and

� .

( ii ) Hence , show that

4 2 � 2 � �� 2 5�

� u �u �1� �u � 2u cos �1��u � 2u cos �1� � 6 �� 6 � ^.

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( iii ) Express of z . z

�z

6 3

� 1 as the product of three quadratic expressions

( iv ) Assume that n is even . Suggest an expression for

2n n z � z �1� 0 as the product of n quadratic expressions of z . [ 9 ]

� 5� 3 ( c ) By using ( b )( ii ), verify that cos cos _{�� .}

6 6 4 [ 3 ]

� 5� 7� ( d ) Find cos cos cos _. 9 9 9

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