Analysis and Approaches for IBDP Maths Ebook 2 | Page 117

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Paper 1 Section A – Forming Polynomials
Example
It is given that z1 �� 5 and z2 �3� 4i are roots of the cubic equation
3 2
z bz cz d
� � � � 0 , where b , c , d � .
(a) Write down the third root z
3
of the equation.
(b) Find the values of b , c and d .
(c)
�4
�
Let � � arctan � �
� 3 � . Express z
3
in the form rcis� and in terms of � and � .
[1]
[4]
[3]
Solution
(a) z3 �3� 4i
A1
(b)
The cubic polynomial
� ( z �( �5))( z �(3� 4i))( z �(3� 4i))
M1
� � � �
2
( z 5)( z 6z
25)
3 2
� z � z �5z
� 125
�b
� � 1, c �� 5 and d � 125
A3
(c) z3 �3�
4i
2 2 � ��4
��
z3
� ( 3 � ( �4) )cis�arctan � ��
� � 3 ��
M1
z3 �5cis(2 � � �)
A2
[1]
[4]
[3]
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