Question

find the remainder obtained by dividing the p(x)-x⁴+x³-2x²+x+1 by (x-1)

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Answer 1

As P(x) is the sum of GP. =

1−x

1−x

6

It has 5 roots , let a

1

,a

2

,a

3

,a

4

,a

5

, and they are the 6th roots of unity except unity.

NowP(x

12

)=1+x

12

+x

24

+x

36

+x

48

+x

60

=P(x).Q(x)+R(x).

Here R(x) is a remainder and a polynomial of maximum degree 4.

Put x=a

1

,a

2

...............,a

5

We get,

R(a

1

)=6, R(a

2

)=6 ,R(a

3

)=6, R(a

4

)=6, R(a

5

)=6

i.e, R(x)−6=0 has 6 roots.

Which contradict that R(x) is maximum of degree 4.

So, it is an identity

Therefore, R(x)=6.