Question

Which of the following statements is correct?

• A. If ${\mathrm{x}}^6+1$ is divisible by $\left(\mathrm{x}+1\right)$, then the remainder is $-2$
• B. If ${\mathrm{x}}^6+1$ is divisible by $\left(\mathrm{x}-1\right)$, then the remainder is $2$
• C. If ${\mathrm{x}}^6+1$ is divisible by $\left(\mathrm{x}+1\right)$, then the remainder is $1$
• D. If ${\mathrm{x}}^6+1$ is divisible by $\left(\mathrm{x}-1\right)$, then the remainder is $-1$

Answer 1

The correct option is B

If ${\mathrm{x}}^6+1$ is divisible by $\left(\mathrm{x}-1\right)$, then the remainder is $2$

Determine the correct answer.

Explanation for the correct answer:

Option(B):

$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^6+1$is divided by $\left(\mathrm{x}-1\right)$.
Since, the remainder can be evaluated using the remainder theorem by substituting the value of the variable in the dividend polynomial.

First equating the divisor ${\mathrm{D}}_1\left(\mathrm{x}\right)=\left(\mathrm{x}-1\right)$ is zero,

So,
$$\begin{gathered} \left(\mathrm{x}-1\right)=0 \\ \Rightarrow \mathrm{x}=1 \end{gathered}$$
substitute the value of $\mathrm{x}$ as $\left(1\right)$ in the original dividend function to get the remainder using the remainder theorem:
$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^6+1 \\ \Rightarrow \mathrm{f}\left(1\right)=1^6+1 \\ \Rightarrow \mathrm{f}\left(1\right)=1+1 \\ \Rightarrow \mathrm{f}\left(1\right)=2 \end{gathered}$$

Hence, option (B) is correct.

Explanation for the incorrect answer:

Option(A), Option(C), and Option(D): the remainder is evaluated using the remainder theorem by substituting the value of the variable in the dividend polynomial, which is $2$.

So, option(A),(C)and(D) are incorrect answers.

Therefore, option(B) is the correct answer.