Question

If ${x^{51}} + 51$ is divided by $x + 1$, Then the remainder is
$
  A)0 \\
  B)1 \\
  C)49 \\
  D)50 \\
 $

Answer 1

Hint: Approach the solution by using remainder theorem concept.
Remainder theorem
 If any polynomial f(x) is divided by f(x-h), then the remainder will be f (h)
Here given polynomial is ${x^{51}} + 51$ and divided by $x + 1$
So by using the remainder theorem we can say that
Remainder=$f( - 1)$ since the divisor is $x + 1$
Therefore we can write it as
$ \Rightarrow f(x) = {x^{51}} + 51 \\
   \Rightarrow f( - 1) = {( - 1)^{51}} + 51 \\
   \Rightarrow f( - 1) = - 1 + 51 \\
  \therefore f( - 1) = 50 \\
    \\
 $
Hence the remainder is $50$
Note: Focus on the divisor of the polynomial which helps to give the remainder value.
Remainder theorem only works when a function is divided by a linear polynomial, which is of the form x + number or x - number.