1 . You are asked to investigate the sum of the powers of roots of a polynomial equation .
Let r 1 and r 2 be the roots of the quadratic equation
2 x a1x a0 0 � � � .
( a ) Express a
1 and a
0 in terms of r 1 and r 2 .
2 2
Let S1 �r1 � r2 and S2 �r1 � r2 defined as the sum of roots and the sum of the squares of roots respectively .
( b ) ( i ) Express a
1 in terms of S
1
.
[ 4 ]
( ii ) Show that a
0
2
S1 � S2
� . 2
Let r 1
, r 2 and r 3 be the roots of the cubic equation x � a x � a x � a � .
3 2 2 1 0
0
[ 4 ]
Let S1 � r1 � r2 � r3 , S � r � r � r and
2 2 2 2 1 2 3
S � r � r � r .
3 3 3 3 1 2 3
( c ) ( i ) Suggest an expression for a
2 in terms of S
1
.
It is given that a1 � rr
1 2
� rr
1 3
� r2 r3 , the sum of roots taken two at a time .
2
S1 � S2 ( ii ) Is it true that a1
� ? Explain your answer . 2
It is also given that ka � S �3S S � 2S
.
[ 5 ]
( d ) By comparing the coefficient of r 1 r 2 r 3 on both sides , find k .
Let r i
( i�
1 , 2 , , n) be the roots of the polynomial equation x � a x � a x � � a x � a � .
n n�1 n�2 n�1 n�2 1 0
0
[ 5 ]
Let S k n
� � r for k � 1 , 2 , , n.
i�1 k i
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