Analysis and Approaches for IBDP Maths Ebook 2 | Page 118

Your Practice Set – Analysis and Approaches for IBDP Mathematics
Exercise 35
1. It is given that z1 �� 2 and z2 � 3� i are roots of the cubic equation
3 2
z b z c z d
� (1 � 3) � (1 � 3) � � 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) Express z
3
in the form
i
re
� .
[1]
[4]
[2]
2. The graph of a polynomial function f( x ) of degree 4 has only two x -intercepts � 5 and
1.
(a) Explain why there are two complex roots for the equation f( x) � 0 .
It is given that
complex root is 3� i.
f x � x � a x �b x � cx � d , where a , b , c , d � . One of the
2
( ) ( )( )( )
(b) Find the expression of f( x ) as a product of two linear factors and a quadratic
factor.
[2]
[4]
4 3 2
3. Consider the quartic equation z �bz � cz � dz � e � 0 , where b , c , d , e� and
z � . Two of the roots of the equation are 4 and 1�
2i . The sum of roots is 3. Show that
b � c � d � e � 49 � 0.
[7]
4 3 2
4. Consider the quartic equation z �bz � cz � dz � e � 0 , where b , c , d , e� and
z � . Two of the roots of the equation are � 3 and 2 � 5i . The product of roots is 174.
Show that b � c � d �15 � 0 .
[7]
108
SE Production Limited