Question

12. Using factor theorem determine whether x-1 is a factor of the following?
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Answer 1

Solution: (i) p(x) = 2x3 + x2 � 2x � 1, g(x) = x + 1

Apply remainder theorem

=>x + 1 =0

=> x = - 1

Replace x by � 1 we get

=>2x3 + x2 � 2x � 1

=>2(-1)3 + (-1)2 -2(-1) - 1

=> -2 + 1 + 2 �- 1

=> 0

Remainder is 0 so that x+1 is a factor of 2x3 + x2 � 2x � 1

(ii) p(x) = x3 + 3x2 + 3x + 1, g(x) = x + 2

Apply remainder theorem

=>x + 2 =0

=> x = - 2

Replace x by � 2 we get

=>x3 + 3x2 + 3x + 1

=>(-2)3 + 3(-2)2 + 3(-2) + 1

=> -8 + 12 - 6 + 1

=> -1

Remainder is not equal to 0 so that x+2 is not a factor of x3 + 3x2 + 3x + 1

(iii) p(x) = x3 � 4x2 + x + 6, g(x) = x � 3

Apply remainder theorem

=>x - 3 =0

=> x = 3

Replace x by � 2 we get

=>x3 � 4x2 + x + 6

=>(3)3 -4(3)2 + 3 + 6

=> 27� - 36�� +3 + 6

=> 0