Question

Which one of the following statements is correct?
A. If ${x^6} + 1$ is divided by $x + 1$ remainder is $ - 2$
B. If ${x^6} + 1$ is divided by $x - 1$ remainder is $2$
C. If ${x^6} + 1$ is divided by $x + 1$ remainder is $1$
D. If ${x^6} + 1$ is divided by $x - 1$ remainder is $ - 1$

Answer 1

Hint: In order to solve this question, we will use factor theorem which states that a polynomial $f(x)$ has a factor $(x - k)$ if and if $f(k) = 0.$ So, by using it we check all the options.

Complete step-by-step answer:
Consider option (A)
By factorization theorem if $(x - k)$ is divided by $f(x)$ then $f(k)$ is remainder.
Here, $f(x) = {x^6} + 1$ which is divided by $x + 1$
So, remainder $f( - 1) = {( - 1)^6} + 1 = 2$
Therefore, option (A) is incorrect.
Consider Option (B)
Here, $f(x) = {x^6} + 1$ which is divided by $x - 1$
So , remainder $f(1) = {(1)^6} + 1 = 2$
Hence, the correct option is “B”.

Note: To solve this type of problem, it is better to start by checking the option. To check if the options are correct or not we use the factorization theorem. This problem can also be solved by a long division method. In that process we have to first divide ${x^6} + 1$ by $x + 1$ and check if the remainder is right or not. If the remainder is not correct then we will divide ${x^6} + 1$ by $x - 1$ and check the answer.