Question

Which of the following statements is correct?
A. If ${x^6} + 1$ is divided by $\left( {x + 1} \right)$, then remainder is $\left( { - 2} \right)$
B. If ${x^6} + 1$ is divided by $\left( {x - 1} \right)$, then remainder is $\left( 2 \right)$
C. If ${x^6} + 1$ is divided by $\left( {x + 1} \right)$, then remainder is $1$
D. If ${x^6} + 1$ is divided by $\left( {x - 1} \right)$, then remainder is $\left( { - 1} \right)$

Answer 1

Hint: In the question, we are provided with a function which is divided by multiple divisor polynomials. The remainder in each case is given to us and we have to choose the correct alternative from the options. We first use the remainder theorem to find the remainders in all the cases. This question requires us to have the knowledge of basic and simple algebraic rules and operations such as substitution, addition, multiplication, subtraction and many more like these. A thorough understanding of functions, division algorithms and its applications will be of great significance.

Complete step by step answer:
In the given question, we have the function $f\left( x \right) = {x^6} + 1$.
Now, the polynomial function is divided by multiple divisor polynomials.
So, we will check the options of the question one by one.

Analyzing option (A):
$f\left( x \right) = {x^6} + 1$ is divided by $\left( {x + 1} \right)$.
Now, we know that we can evaluate the remainder using the remainder theorem by substituting the value of the variable in the dividend polynomial. First equating the divisor ${D_1}\left( x \right) = \left( {x + 1} \right)$ as zero, we get the value of $x$ as,
$ \Rightarrow \left( {x + 1} \right) = 0$
$ \Rightarrow x = - 1$
So, we substitute the value of x as $\left( { - 1} \right)$ in the original dividend function to get the remainder using the remainder theorem.
So, $f\left( { - 1} \right) = {\left( { - 1} \right)^6} + 1$
So, we compute the powers of $\left( { - 1} \right)$.
$ \Rightarrow f\left( { - 1} \right) = 1 + 1$
Opening the brackets and simplifying the expression,
$ \Rightarrow f\left( { - 1} \right) = 2$
So, the remainder when $f\left( x \right) = {x^6} + 1$ is divided by $\left( {x + 1} \right)$ is $2$. Hence, option (A) is incorrect.

Analyzing option (B):
$f\left( x \right) = {x^6} + 1$is divided by $\left( {x - 1} \right)$.
Now, we know that we can evaluate the remainder using the remainder theorem by substituting the value of the variable in the dividend polynomial. First equating the divisor ${D_1}\left( x \right) = \left( {x - 1} \right)$ as zero, we get the value of $x$ as,
$ \Rightarrow \left( {x - 1} \right) = 0$
$ \Rightarrow x = 1$
So, we substitute the value of x as $\left( 1 \right)$ in the original dividend function to get the remainder using the remainder theorem.
So, $f\left( 1 \right) = {1^6} + 1$
So, we compute the powers of $\left( 1 \right)$.
$ \Rightarrow f\left( 1 \right) = 1 + 1$
Adding up the terms,
$ \Rightarrow f\left( 1 \right) = 2$
So, the remainder when $f\left( x \right) = {x^6} + 1$is divided by $\left( {x - 1} \right)$ is $2$. Hence, option (B) is correct.

Analyzing option (C):
$f\left( x \right) = {x^6} + 1$is divided by $\left( {x + 1} \right)$.
Now, we know that the remainder when $f\left( x \right) = {x^6} + 1$is divided by $\left( {x + 1} \right)$ is $2$.
Hence, option (C) is incorrect.

Analyzing option (D):
$f\left( x \right) = {x^6} + 1$is divided by $\left( {x - 1} \right)$.
Now, we also know that the remainder when $f\left( x \right) = {x^6} + 1$is divided by $\left( {x - 1} \right)$ is $2$.
Hence, option (D) is incorrect.

Therefore, option B is the correct answer.

Note: Remainder theorem is an approach of Euclidean division of polynomials.Remainder theorem requires just a simple change of variable in the function so as to find the remainder of the division procedure. Substitution of a variable involves putting a certain value in place of the variable. That specified value may be a certain number or even any other variable. We must take care of the calculations in order to be sure of the final answer.