If f( x)=x^ 4 -2x^ 3 +3x^ 2 -9x+1 b is a polynomial such that when it is divided by x - 1 and x + 1 . the remainders are respectively 5 and 19. Determine the remainder when I(x, ) is divided by (x-2)
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The remainder when p is divided by x−1 is is 6 , therefore (remainder theorem)
p(1)=6 .
The other condition, can be written
p(−1)=14 .
These equations are written in terms of a and b as
1−2+3−a+b=6
1+2+3+a+b=14 .
or
−a+b=4
a+b=8 ,
and hence
a=2
b=6 .
The remainder when p is divided by x−2 is (remainder theorem again)
p(2)=16−16+12−4+6=14 .
Therefore, the remainder is 14.
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