Analysis and Approaches for IBDP Maths Ebook 2 | Page 33

7
Example
Paper 1 Section A – Factor Theorem
2
3 2
The polynomial P( x) � ax � 6x �bx
� 12 is divisible by ( x � 1) and by ( x � 4) . Find the
value of a and the value of b , where a , b� .
[5]
Solution
P(1) � 0
a
3 2
(1) 6(1) b(1) 12 0
� � � � (M1) for factor theorem
b�6
� a
A1
P( �4) � 0
a
3 2
( �4) � 6( �4) �b( �4) �12 � 0
�64a� 96 � 4b�12 � 0
��64a� 96 � 4(6 � a) �12 � 0
(M1) for substitution
�64a� 96 � 24 � 4a�12 � 0
�60a
� � 60
a � 1
A1
b �6�
1
b � 5
A1
[5]
Exercise 7
1. The polynomial
3 2
f ( x) � ax �bx �13x
� 6 is divisible by ( x � 2) and by ( x � 3) . Find the
value of a and the value of b , where a , b� .
2. The cubic polynomial
3 2
x px qx
� � � 48 has a factor ( x � 4) and leaves a
remainder 105 when divided by ( x � 3) . Find the value of p and the value of q , where
p , q� .
[5]
[5]
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