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Last updated date: 07th Dec 2023
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# What is the remainder when $\left( {{x}^{11}}+1 \right)$ is divided by $\left( x+1 \right)$?A.0B.2C.11D.12

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Hint: In this problem, we have to find the remainder, when $\left( {{x}^{11}}+1 \right)$ is divided by $\left( x+1 \right)$. Here we can see that the polynomial expression has the highest power, so we can use the method remainder theorem concept and find the remainder. If any polynomial $f\left( x \right)$ is divided by $f\left( x-h \right)$, then the remainder will be $f\left( h \right)$, we can now use this theorem to find the remainder.

Here we have to find the remainder, when $\left( {{x}^{11}}+1 \right)$ is divided by $\left( x+1 \right)$.
Here we have high power polynomial expression, so we can use the remainder theorem concept.
We know that, If any polynomial $f\left( x \right)$ is divided by $f\left( x-h \right)$, then the remainder will be $f\left( h \right)$.
So, by using the remainder theorem we can say that
Since, $\left( x+1 \right)$ is the divisor,
Then the remainder is
$\Rightarrow \operatorname{R}=f\left( -1 \right)$
We can now write as,
\begin{align} & \Rightarrow f\left( x \right)={{x}^{11}}+1 \\ & \Rightarrow f\left( -1 \right)={{\left( -1 \right)}^{11}}+1=-1+1=0 \\ \end{align}
Hence, the remainder is 0 when $\left( {{x}^{11}}+1 \right)$ is divided by $\left( x+1 \right)$.

So, the correct answer is “Option A”.

Note: We should always focus on the divisor of the polynomial which helps to give the remainder value. We should also know that the remainder theorem only works when a function is divided by a linear polynomial, which is of the form x+ number or x- number. Here we have not used the polynomial long division or synthetic division methods as the given polynomial expression has the highest power raised to it.