If the polynomial $x^{19}+x^{17}+x^{13}+x^{11}+x^{7}+x^{5}+x^{3}$ is divided by $( x^{2}-1)$, then find the remainder.

Given: Polynomial $x^{19}+x^{17}+x^{13}+x^{11}+x^{7}+x^{5}+x^{3}$ is divided by $( x^{2}-1)$.

To do: To find the remainder.

Solution:

As given, polynomial $x^{19}+x^{17}+x^{13}+x^{11}+x^{7}+x^{5}+x^{3}$ is divided by $( x^{2}-1)$.

Let $x^2-2=0$

$\Rightarrow x^2=1$

$\Rightarrow x=\pm1$, Put this value in the given [polynomial:

Remainder$=1^{19}+1^{17}+1^{13}+1^{11}+1^{7}+1^{5}+1^{3}$

$=1+1+1+1+1+1+1$                      [$\because 1^n=1$]

$=7$

Thus, the remainder is $7$.

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Updated on: 10-Oct-2022

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