Question

If $(x^{31}+31)$ is divided by $\left(x+1\right)$, then find the remainder.

Answer 1

Compute the required value.

As we know, if a polynomial function $p\left(x\right)$ is divided by $\left(x-a\right)$, then the reminder $R$ is given by:

$R=p\left(a\right)$

Assume that, $p\left(x\right)=x^{31}+1$.

Since $p\left(x\right)$ is divided by $\left(x+1\right)$.

Therefore, $R=p(-1)$.

$$\begin{gathered} \Rightarrow R={(-1)}^{31}+31 \\ \Rightarrow R=-1+31 \\ \Rightarrow R=30 \end{gathered}$$

Hence, the value of the remainder is $30$.