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P ( x ) = ( x999 + x998 + x997 + . . . . + x + 1)2 - x999Q ( x ) = x998 + x 997 + . . . . + x + 1The reminder when P ( x ) is divided by Q ( x ) is
Ax + 1
B0
C1
Dx-1
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Solution: ((x^999 x^998 ... x 1)^2-x^999)/(x^998 x^997 ... x 1)
=> ((x^999 x^998 ... x 1)^2-x^999)/(x^999 x^998 ... x 1)-x^999 or,
=>(x^998 ... x 2)^2-1/(x^998 ... x 2)-1 [taking x^999 as common and cancelling it]
we can now use the identity a^2-b^2=(a b)*(a-b)
using this we can see that p(x) is clearly divisible by q(x)
i.e. p(x)/q(x)= (x^999 x^998 ... x 1) x^999
so the remainder will be 0
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Solution: ((x^999 x^998 ... x 1)^2-x^999)/(x^998 x^997 ... x 1)
=> ((x^999 x^998 ... x 1)^2-x^999)/(x^999 x^998 ... x 1)-x^999 or,
=>(x^998 ... x 2)^2-1/(x^998 ... x 2)-1 [taking x^999 as common and cancelling it]
we can now use the identity a^2-b^2=(a b)*(a-b)
using this we can see that p(x) is clearly divisible by q(x)
i.e. p(x)/q(x)= (x^999 x^998 ... x 1) x^999
so the remainder will be 0
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