Question

The remainder obtained when the polynomial ${x^{64}} + {x^{27}} + 1$ is divided by $\left( {x + 1} \right)$ is:
A. $1$
B. $ - 1$
C. $2$
D. $ - 2$

Answer 1

Hint: In the question, we are provided with a function which is divided by a divisor polynomial. We have to evaluate the remainder. We use the remainder theorem to find the remainder. We will first equate the divisor to zero in order to obtain the value of variable x. Then, we substitute the value of variable x obtained in the polynomial to find out the remainder using the remainder theorem. 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.

Formula used:
Remainder theorem: When $p\left( x \right)$ is divided by $\left( {x - a} \right)$, the remainder obtained is $r = p\left( a \right)$.

Complete step by step solution:
In the given question, we are given the function $f\left( x \right) = {x^{64}} + {x^{27}} + 1$.
Now, the polynomial function is divided by the divisor polynomial $\left( {x + 1} \right)$.
Now, we know that we can evaluate the remainder using the remainder theorem by substituting the value of the variable in the dividend.
According to the remainder theorem, the remainder obtained on dividing the polynomial $p\left( x \right)$ by the divisor $\left( {x - a} \right)$ is given by $r = p\left( a \right)$. So, we first equate the divisor ${D_1}\left( x \right) = \left( {x + 1} \right)$ as zero.
So, we get,
$ \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, $f\left( { - 1} \right) = {\left( { - 1} \right)^{64}} + {\left( { - 1} \right)^{27}} + 1$
So, we compute the powers of $\left( { - 1} \right)$.
$ \Rightarrow f\left( { - 1} \right) = 1 + \left( { - 1} \right) + 1$
Opening the brackets and simplifying the expression,
$ \Rightarrow f\left( { - 1} \right) = 1 - 1 + 1$
$ \Rightarrow f\left( { - 1} \right) = 1$

So, (A) 1 is the remainder obtained when the polynomial ${x^{64}} + {x^{27}} + 1$ is divided by $\left( {x + 1} \right)$.

Hence option (A) is correct.

Note: Remainder theorem is a Euclidean approach of 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.